C₂×#33962 #7623950

order	512
nilpotency class	4
isoclinism	isoclinic class of D₄×D₁₆
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

Aliases

direct product

C₂×#33962

Subgroups

	subgroup	G/H
center, socle, prime radical	C₂³	C₂²×D₈
commutator, Frattini, cosocle	C₂×C₈	C₂⁵
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄×Q₈ ◁ C₂×C₈×Q₈ ◁ G
LECS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁

1370 conjugacy classes of subgroups; only the 562 normal subgroups are shown.

normal subgroups only

This group has 1370 conjugacy classes of subgroups — too many to draw. Only its 562 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 — 5 generators on 42 points:

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

Conjugacy classes & character table

110 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7531146);