C₂×#132069 #72292758

order	1024
nilpotency class	4
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁷

Aliases

direct product

C₂×#132069

Subgroups

	subgroup	G/H
center, socle	C₂²	#8456
commutator	#117	C₂³×C₄
Frattini, cosocle	#405	C₂⁴
prime radical	C₂×#62270	C₂
LCS	G ▷ #117 ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×#405 ◁ G
LECS	G ▷ #405 ▷ C₂×C₄ ▷ C₂ ▷ C₁

4158 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:

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

Conjugacy classes & character table

88 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#72292716
maximal normal subgroup of	C₂×#72292716

GAP

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

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

C2×#132069 #72292758