C₂×#12469 #490563

order	512
nilpotency class	4
isoclinism	isoclinic class of #12430
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

direct product

C₂×#12469

Subgroups

	subgroup	G/H
center, socle	C₂³	#466
commutator, Frattini, cosocle	C₄×C₈	C₂⁴
prime radical	C₂²×D₄	D₈
LCS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₈×D₄ ◁ G
LECS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

502 conjugacy classes of subgroups; only the 162 normal subgroups are shown.

normal subgroups only

This group has 502 conjugacy classes of subgroups — too many to draw. Only its 162 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) = 256

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

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

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, 397759);