C₂×#12327 #488910

order	512
nilpotency class	4
isoclinism	isoclinic class of #12324
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

direct product

C₂×#12327

Subgroups

	subgroup	G/H
center, socle	C₂³	#447
commutator, Frattini, cosocle	C₄×C₈	C₂⁴
prime radical	C₂×#3277	C₂
LCS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×#436 ◁ G
LECS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

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

normal subgroups only

This group has 991 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)(5,8)(6,9)(7,10)(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);(4,8)(5,6)(9,10)(11,13)(12,18)(14,23)(15,28)(16,32)(17,25)(19,34)(20,37)(21,35)(22,27)(24,29)(26,33)(30,42)(31,36)(38,40)(39,41);(4,9)(5,6)(8,10)(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, 396106);