C₂×#12351 #489011

order	512
nilpotency class	4
isoclinism	isoclinic class of #12324
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

direct product

C₂×#12351

Subgroups

	subgroup	G/H
center, socle	C₂³	#447
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×#436 ◁ G
LECS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

727 conjugacy classes of subgroups; only the 154 normal subgroups are shown.

normal subgroups only

This group has 727 conjugacy classes of subgroups — too many to draw. Only its 154 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);(4,5)(6,7)(9,10)(11,12,17,21)(13,22,25,40)(14,19,16,31)(15,27,30,38)(18,33,35,24)(20,26,39,29)(23,41,32,37)(28,34,42,36);(3,4,8,10)(5,6,9,7)(11,13,17,25)(12,18,21,35)(14,23,16,32)(15,28,30,42)(19,34,31,36)(20,37,39,41)(22,38,40,27)(24,26,33,29);(3,5)(4,6)(7,10)(8,9)(11,14,17,16)(12,19,21,31)(13,23,25,32)(15,29,30,26)(18,34,35,36)(20,38,39,27)(22,37,40,41)(24,42,33,28)

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 396207);