C₂×#152133 #72256451

order	1024
nilpotency class	5
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁸

Aliases

direct product

C₂×#152133

Subgroups

	subgroup	G/H
center, socle	C₂²	#12707
commutator	#117	C₂³×C₄
Frattini, cosocle	#535	C₂⁴
prime radical	C₂×#61630	C₂
LCS	G ▷ #117 ▷ C₂×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³ ◁ C₂³×C₄ ◁ C₂×#3484 ◁ G
LECS	G ▷ #535 ▷ C₂×C₄ ▷ C₂² ▷ C₂ ▷ C₁

5127 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:

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

Conjugacy classes & character table

76 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#72216447, C₂×#72292695
maximal normal subgroup of	C₂×#72216447, C₂×#72292695
maximal quotient of	#72293108

GAP

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

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

C2×#152133 #72256451