C₂×#12345 #488986

order	512
nilpotency class	4
isoclinism	isoclinic class of #12316
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

direct product

C₂×#12345

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×C₈×D₄ ◁ G
LECS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

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

normal subgroups only

This group has 736 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);(4,5)(6,7)(9,10)(11,12,17,21)(13,22,25,40)(14,19,16,31)(15,27,30,41)(18,33,36,24)(20,26,38,29)(23,39,32,37)(28,34,42,35);(3,4,8,10)(5,6,9,7)(11,13)(12,18)(14,23)(15,28)(16,32)(17,25)(19,34)(20,37)(21,36)(22,27)(24,29)(26,33)(30,42)(31,35)(38,39)(40,41);(3,5)(4,6)(7,10)(8,9)(11,14,17,16)(12,19,21,31)(13,23,25,32)(15,26,30,29)(18,34,36,35)(20,27,38,41)(22,39,40,37)(24,28,33,42)

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 396182);