C₂×#7393 #150079

order	512
nilpotency class	3
isoclinism	isoclinic class of #327
automorphism	?
homology	Abelianization (H₁) = C₂²×C₁₆ · Schur multiplier (H₂) = C₂²

Aliases

direct product

C₂×#7393

Subgroups

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

224 conjugacy classes of subgroups; only the 114 normal subgroups are shown.

normal subgroups only

This group has 224 conjugacy classes of subgroups — too many to draw. Only its 114 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)(11,12,14,20,16,21,27,35,17,18,25,30,29,31,38,40,15,19,26,32,28,33,39,41,13,22,23,34,24,36,37,42);(3,4,6,8)(5,10,9,7)(11,13,15,17)(12,18,19,22)(14,25,26,23)(16,24,28,29)(20,34,32,30)(21,31,33,36)(27,38,39,37)(35,42,41,40)

Conjugacy classes & character table

152 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 57275);