C₂×#12530 #491507

order	512
nilpotency class	4
isoclinism	isoclinic class of #12525
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₈

Aliases

direct product

C₂×#12530

Subgroups

	subgroup	G/H
center, socle	C₂³	#493
commutator, Frattini, cosocle	C₄×C₈	C₂⁴
prime radical	C₂×#493	C₂²
LCS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₈² ◁ G
LECS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

664 conjugacy classes of subgroups; only the 186 normal subgroups are shown.

normal subgroups only

This group has 664 conjugacy classes of subgroups — too many to draw. Only its 186 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);(4,7)(5,9)(8,10)(11,12,14,17)(13,21,23,35)(15,28,26,41)(16,29,27,20)(18,32,30,24)(19,34,31,22)(25,40,39,37)(33,38,36,42);(3,4,5,7,6,8,9,10)(12,17)(18,30)(19,31)(20,29)(21,35)(28,41)(33,36)(37,40);(4,8)(7,10)(11,13,22,38,16,25,26,32,14,23,34,42,27,39,15,24)(12,18,28,37,20,33,31,35,17,30,41,40,29,36,19,21)

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 398703);