C₂×#12507 #491434

order	512
nilpotency class	4
isoclinism	isoclinic class of #12506
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

direct product

C₂×#12507

Subgroups

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

594 conjugacy classes of subgroups; only the 202 normal subgroups are shown.

normal subgroups only

This group has 594 conjugacy classes of subgroups — too many to draw. Only its 202 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 — 4 generators on 42 points:

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

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, 398630);