C₂×#151129 #72231997

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

Aliases

direct product

C₂×#151129

Subgroups

	subgroup	G/H
center, socle	C₂³	#2177
commutator	C₂×C₁₆	C₂³×C₄
Frattini, cosocle	C₄×C₁₆	C₂⁴
prime radical	C₂²×D₃₂	C₄
LCS	G ▷ C₂×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₈² ◁ C₂×#1311 ◁ G
LECS	G ▷ C₄×C₁₆ ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁

657 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 42
transitive degree	μ_t(G) = 512

A minimal faithful permutation representation — 4 generators on 42 points:

(1,2);(3,4,7,6)(5,9,10,8)(12,17)(13,22)(14,24)(15,26)(18,28)(19,23)(20,30)(21,37)(25,33)(27,40)(29,31)(32,34)(35,42)(36,41)(38,39);(3,5,6,9,7,10,4,8)(12,18,20,34)(13,23)(17,28,30,32)(19,22)(21,29,36,42)(25,38)(31,41,35,37)(33,39);(11,12,19,35,14,21,25,32,26,34,39,37,40,42,13,17,16,20,22,31,27,36,38,28,15,18,33,41,24,29,23,30)

Conjugacy classes & character table

112 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#72216311
maximal normal subgroup of	C₂×#72216311

GAP

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

gap> G := Group((1,2), (3,4,7,6)(5,9,10,8)(12,17)(13,22)(14,24)(15,26)(18,28)(19,23)(20,30)(21,37)(25,33)(27,40)(29,31)(32,34)(35,42)(36,41)(38,39), (3,5,6,9,7,10,4,8)(12,18,20,34)(13,23)(17,28,30,32)(19,22)(21,29,36,42)(25,38)(31,41,35,37)(33,39), (11,12,19,35,14,21,25,32,26,34,39,37,40,42,13,17,16,20,22,31,27,36,38,28,15,18,33,41,24,29,23,30));

C2×#151129 #72231997