C₂×#12339 #489191

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

Aliases

direct product

C₂×#12339

Subgroups

	subgroup	G/H
center, socle	C₂³	#447
commutator, Frattini, cosocle	C₄×C₈	C₂⁴
prime radical	C₂²×D₈	D₄
LCS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×#446 ◁ G
LECS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

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

normal subgroups only

This group has 785 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);(4,7)(5,9)(8,10)(11,12,16,20)(13,21,23,39)(14,25,27,41)(15,30,28,19)(17,32,33,22)(18,35,37,29)(24,40,31,36)(26,38,42,34);(5,9)(7,10)(11,13)(12,17)(14,26)(15,31)(16,23)(18,36)(19,38)(20,33)(21,25)(22,29)(24,28)(27,42)(30,34)(32,35)(37,40)(39,41);(3,4,6,8)(5,10,9,7)(14,27)(18,37)(21,39)(22,32)(25,41)(26,42)(29,35)(36,40)

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, 396387);