C₂²×#12666 #72313464

order	1024
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄² · Schur multiplier (H₂) = C₂⁹

Aliases

direct product

C₂²×#12666

Subgroups

	subgroup	G/H
center	C₂³×C₄	C₂×C₄²
commutator	C₂×C₄	C₂³×C₄²
Frattini, cosocle	C₂²×C₈	C₂⁵
socle	C₂⁴	C₂×#97
prime radical	C₂⁵	C₂×M₁₆
LCS	G ▷ C₂×C₄ ▷ C₁
UCS	C₁ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

4749 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 24
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 24 points:

(6,8)(10,14,16,22)(11,18)(15,20)(17,24,19,23);(1,2);(3,4);(1,2)(3,4)(5,6,7,8);(1,2)(3,4)(9,10,15,23,21,22,18,17,13,16,20,24,12,14,11,19)

Conjugacy classes & character table

256 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂²×#498331, C₂²×#497864, C₂²×#498334, …
maximal normal subgroup of	C₂²×#498331, C₂²×#497864, C₂²×#498334, …

GAP

Past the SmallGroups library coverage — no SmallGroups id. Construct it from a minimal faithful permutation representation:

gap> G := Group((6,8)(10,14,16,22)(11,18)(15,20)(17,24,19,23), (1,2), (3,4), (1,2)(3,4)(5,6,7,8), (1,2)(3,4)(9,10,15,23,21,22,18,17,13,16,20,24,12,14,11,19));

C22×#12666 #72313464