C₂×#7451 #150952

order	512
nilpotency class	5
isoclinism	isoclinic class of #7418
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂³

Aliases

direct product

C₂×#7451

Subgroups

	subgroup	G/H
center, socle, prime radical	C₂³	#357
commutator	C₂×C₁₆	C₂²×C₄
Frattini, cosocle	C₄×C₁₆	C₂³
LCS	G ▷ C₂×C₁₆ ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ C₂×C₄×C₁₆ ◁ G
LECS	G ▷ C₄×C₁₆ ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁

222 conjugacy classes of subgroups; only the 106 normal subgroups are shown.

normal subgroups only

This group has 222 conjugacy classes of subgroups — too many to draw. Only its 106 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 3 generators on 42 points:

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

Conjugacy classes & character table

98 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 58148);