C₂×#129615 #72307123

order	1024
nilpotency class	4
automorphism	?
homology	Abelianization (H₁) = C₂³×C₈ · Schur multiplier (H₂) = C₂⁸

Aliases

direct product

C₂×#129615

Subgroups

	subgroup	G/H
center	C₂×C₈	B(4,𝔽₂)
commutator	C₂×D₄	C₂³×C₈
Frattini, cosocle	C₂×#131	C₂⁴
socle	C₂²	#8316
prime radical	C₂×#3481	C₄
LCS	G ▷ C₂×D₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₈ ◁ C₂²×C₈ ◁ C₂²×#131 ◁ G
LECS	G ▷ C₂×#131 ▷ C₂×C₄ ▷ C₂ ▷ C₁

4386 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 34
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 4 generators on 34 points:

(9,13)(21,26)(22,27)(32,34);(1,2);(4,9,11,13)(10,21,23,26)(12,22,25,27)(24,32,33,34);(3,4,6,10,7,12,17,24,8,11,18,23,19,25,30,33)(5,13,14,26,16,27,28,34,15,9,20,21,29,22,31,32)

Conjugacy classes & character table

160 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72307130, #72307167, #72307601, …
maximal normal subgroup of	#72307130, #72307167, #72307601, …

GAP

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

gap> G := Group((9,13)(21,26)(22,27)(32,34), (1,2), (4,9,11,13)(10,21,23,26)(12,22,25,27)(24,32,33,34), (3,4,6,10,7,12,17,24,8,11,18,23,19,25,30,33)(5,13,14,26,16,27,28,34,15,9,20,21,29,22,31,32));

C2×#129615 #72307123