Group cohomology of groups of order 8

This article gives specific information, namely, group cohomology, about a family of groups, namely: groups of order 8.
View group cohomology of group families | View group cohomology of groups of a particular order |View other specific information about groups of order 8
Group GAP ID 2nd part Hall-Senior number Nilpotency class If it is a finite group with periodic cohomology, the period for the homology groups Cohomology information page
cyclic group:Z8 1 3 1 2 group cohomology of cyclic group:Z8, see also group cohomology of finite cyclic groups
direct product of Z4 and Z2 2 2 1 -- group cohomology of direct product of Z4 and Z2
dihedral group:D8 3 4 2 -- group cohomology of dihedral group:D8
quaternion group 4 5 2 4 group cohomology of quaternion group
elementary abelian group:E8 5 1 1 -- group cohomology of elementary abelian group:E8

With the exception of the zeroth homology group and cohomology group, all homology and cohomology groups over all possible abelian groups are 2-groups.

Homology groups for trivial group action

FACTS TO CHECK AGAINST (homology group for trivial group action):
First homology group: first homology group for trivial group action equals tensor product with abelianization
Second homology group: formula for second homology group for trivial group action in terms of Schur multiplier and abelianization|Hopf's formula for Schur multiplier
General: universal coefficients theorem for group homology|homology group for trivial group action commutes with direct product in second coordinate|Kunneth formula for group homology

Over the integers

The table below lists the first few homology groups with coefficients in the integers. We use $\mathbb{Z}_n$ to denote the cyclic group of order $n$.

We use 0 to denote the trivial group.

Group GAP ID 2nd part Hall-Senior number Nilpotency class $H_1$ (= abelianization) $H_2$ (= Schur multiplier) $H_3$ $H_4$ $H_5$
cyclic group:Z8 1 3 1 $\mathbb{Z}_8$ 0 $\mathbb{Z}_8$ 0 $\mathbb{Z}_8$
direct product of Z4 and Z2 2 2 1 $\mathbb{Z}_4 \oplus \mathbb{Z}_2$ $\mathbb{Z}_2$ $\mathbb{Z}_4 \oplus (\mathbb{Z}_2)^2$ $(\mathbb{Z}_2)^2$ $\mathbb{Z}_4 \oplus (\mathbb{Z}_2)^3$
dihedral group:D8 3 4 2 $\mathbb{Z}_2 \oplus \mathbb{Z}_2$ $\mathbb{Z}_2$ $\mathbb{Z}_4 \oplus (\mathbb{Z}_2)^2$ $\mathbb{Z}_2 \oplus \mathbb{Z}_2$ $(\mathbb{Z}_2)^4$
quaternion group 4 5 2 $\mathbb{Z}_2 \oplus \mathbb{Z}_2$ 0 $\mathbb{Z}_8$ 0 $\mathbb{Z}_2 \oplus \mathbb{Z}_2$
elementary abelian group:E8 5 1 1 $(\mathbb{Z}_2)^3$ $(\mathbb{Z}_2)^3$ $(\mathbb{Z}_2)^7$ $(\mathbb{Z}_2)^8$ $(\mathbb{Z}_2)^{13}$

Over an abelian group

Below are the homology groups for trivial group action with coefficients in an abelian group $M$. We denote by $aM$ the subgroup $\{ am \mid m \in M \}$ and by $\operatorname{Ann}_M(a)$ the subgroup $\{ x \in M \mid ax = 0 \}$.

These groups can be computed from the homology groups with coefficients in the integers by using the universal coefficients theorem for group homology.

Group GAP ID 2nd part Hall-Senior number Nilpotency class $H_1$ $H_2$ $H_3$ $H_4$ $H_5$
cyclic group:Z8 1 3 1 $M/8M$ $\operatorname{Ann}_M(8)$ $M/8M$ $\operatorname{Ann}_M(8)$ $M/8M$
direct product of Z4 and Z2 2 2 1 $M/4M \oplus M/2M$ $M/2M \oplus \operatorname{Ann}_M(4) \oplus \operatorname{Ann}_M(2)$ $M/4M \oplus (M/2M)^2 \oplus \operatorname{Ann}_M(2)$ $(M/2M)^2 \oplus \operatorname{Ann}_M(4) \oplus (\operatorname{Ann}_M(2))^2$ $M/4M \oplus (M/2M)^3 \oplus \operatorname{Ann}_M(2)$
dihedral group:D8 3 4 2 $M/2M \oplus M/2M$ $M/2M \oplus \operatorname{Ann}_M(2) \oplus \operatorname{Ann}_M(2)$ $M/2M \oplus M/2M \oplus M/4M \oplus \operatorname{Ann}_M(2)$ $M/2M \oplus M/2M \oplus \operatorname{Ann}_M(2) \oplus \operatorname{Ann}_M(2) \oplus \operatorname{Ann}_M(4)$ $(M/2M)^4 \oplus (\operatorname{Ann}_M(2))^2$
quaternion group 4 5 2 $M/2M \oplus M/2M$ $\operatorname{Ann}_M(2) \oplus \operatorname{Ann}_M(2)$ $M/8M$ $\operatorname{Ann}_M(8)$ $M/2M \oplus M/2M$
elementary abelian group:E8 5 1 1 $(M/2M)^3$ $(M/2M)^3 \oplus (\operatorname{Ann}_M(2))^3$ $(M/2M)^7 \oplus (\operatorname{Ann}_M(2))^3$ $(M/2M)^8 \oplus (\operatorname{Ann}_M(2))^7$ $(M/2M)^{13} \oplus (\operatorname{Ann}_M(2))^8$

Cohomology groups for trivial group action

FACTS TO CHECK AGAINST (cohomology group for trivial group action):
First cohomology group: first cohomology group for trivial group action is naturally isomorphic to group of homomorphisms
Second cohomology group: formula for second cohomology group for trivial group action in terms of Schur multiplier and abelianization
In general: dual universal coefficients theorem for group cohomology relating cohomology with arbitrary coefficientsto homology with coefficients in the integers. |Cohomology group for trivial group action commutes with direct product in second coordinate | Kunneth formula for group cohomology

Over the integers

The table below lists the first few cohomology groups with coefficients in the integers. We use $\mathbb{Z}_n$ to denote the cyclic group of order $n$. Note that each cohomology group is just the previous homology group, i.e., $H^q \cong H_{q-1}$. This is a consequence of the dual universal coefficients theorem for group cohomology.

All the $H^1$ groups are trivial groups. We use 0 to denote the trivial group.

Group GAP ID 2nd part Hall-Senior number Nilpotency class $H^1$ $H^2$ $H^3$ $H^4$ $H^5$
cyclic group:Z8 1 3 1 0 $\mathbb{Z}_8$ 0 $\mathbb{Z}_8$ 0
direct product of Z4 and Z2 2 2 1 0 $\mathbb{Z}_4 \oplus \mathbb{Z}_2$ $\mathbb{Z}_2$ $\mathbb{Z}_4 \oplus (\mathbb{Z}_2)^2$ $(\mathbb{Z}_2)^2$
dihedral group:D8 3 4 2 0 $\mathbb{Z}_2 \oplus \mathbb{Z}_2$  ?  ?  ?
quaternion group 4 5 2 0 $\mathbb{Z}_2 \oplus \mathbb{Z}_2$  ?  ?  ?
elementary abelian group:E8 5 1 1 0 $(\mathbb{Z}_2)^3$ $(\mathbb{Z}_2)^3$ $(\mathbb{Z}_2)^7$ $(\mathbb{Z}_2)^8$

Over an abelian group

Group GAP ID 2nd part Hall-Senior number Nilpotency class $H^1$ $H^2$ $H^3$ $H^4$ $H^5$
cyclic group:Z8 1 3 1 $\operatorname{Ann}_M(8)$ $M/8M$ $\operatorname{Ann}_M(8)$ $M/8M$ $\operatorname{Ann}_M(8)$
direct product of Z4 and Z2 2 2 1 $\operatorname{Ann}_M(4) \oplus \operatorname{Ann}_M(2)$ $\operatorname{Ann}_M(2) \oplus M/4M \oplus M/2M$ $\operatorname{Ann}_M(4) \oplus (\operatorname{Ann}_M(2))^2 \oplus M/2M$ $(\operatorname{Ann}_M(2))^2 \oplus M/4M \oplus (M/2M)^2$ $\operatorname{Ann}_M(4) \oplus (\operatorname{Ann}_M(2))^3 \oplus M/2M$
dihedral group:D8 3 4 2 $\operatorname{Ann}_M(2) \oplus \operatorname{Ann}_M(2)$ $\operatorname{Ann}_M(2) \oplus M/2M \oplus M/2M$ $\operatorname{Ann}_M(4) \oplus (\operatorname{Ann}_M(2))^2 \oplus M/2M$ $(\operatorname{Ann}_M(2))^2 \oplus M/4M \oplus (M/2M)^2$ $(\operatorname{Ann}_M(2))^4 \oplus (M/2M)^2$
quaternion group 4 5 2 $\operatorname{Ann}_M(2) \oplus \operatorname{Ann}_M(2)$ $M/2M \oplus M/2M$  ?  ?  ?
elementary abelian group:E8 5 1 1 $(\operatorname{Ann}_M(2))^3$ $(\operatorname{Ann}_M(2))^3 \oplus (M/2M)^3$ $(\operatorname{Ann}_M(2))^7 \oplus (M/2M)^3$ $(\operatorname{Ann}_M(2))^8 \oplus (M/2M)^7$ $(\operatorname{Ann}_M(2))^{13} \oplus (M/2M)^8$

Second cohomology groups and extensions

Schur multiplier and Schur covering groups

FACTS TO CHECK AGAINST for Schur multiplier of group of prime power order
ABELIAN CASE: Schur multiplier of finite abelian group is its exterior square
UPPER BOUNDS: upper bound on size of Schur multiplier for group of prime power order based on prime-base logarithm of order |upper bound on size of Schur multiplier for group of prime power order based on prime-base logarithm of order and exponent of center
LOWER BOUNDS: lower bound on size of Schur multiplier for group of prime power order based on minimum size of generating set

The Schur multiplier is defined as second cohomology group for trivial group action, $H^2(G;\mathbb{C}^\ast)$, and also as the second homology group $H_2(G;\mathbb{Z})$. A corresponding Schur covering group of $G$ is a group that arises as a stem extension with base normal subgroup the Schur multiplier and the quotient group is $G$.

Group GAP ID 2nd part Hall-Senior number Nilpotency class $H_2$ (= Schur multiplier) Order of $H_2$ Possibilities for Schur covering groups Cohomology group information Orders of Schur covering groups exterior square, which is the derived subgroup of every Schur covering group quotient group by epicenter, which is the inner automorphism group of every Schur covering group
cyclic group:Z8 1 3 1 trivial group 1 cyclic group:Z8 -- 8 trivial group trivial group
direct product of Z4 and Z2 2 2 1 cyclic group:Z2 2 SmallGroup(16,3), nontrivial semidirect product of Z4 and Z4, M16 second cohomology group for trivial group action of direct product of Z4 and Z2 on Z2 16 cyclic group:Z2 Klein four-group
dihedral group:D8 3 4 2 cyclic group:Z2 2 dihedral group:D16, semidihedral group:SD16, generalized quaternion group:Q16 second cohomology group for trivial group action of D8 on Z2 16 cyclic group:Z4 dihedral group:D8
quaternion group 4 5 2 trivial group 1 quaternion group -- 8 cyclic group:Z2 Klein four-group
elementary abelian group:E8 5 1 1 elementary abelian group:E8 8 lots of them second cohomology group for trivial group action of E8 on E8 64 elementary abelian group:E8 elementary abelian group:E8