C₂×#30234 #6436340

order	512
nilpotency class	3
isoclinism	isoclinic class of #29742
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁵×C₄²

Aliases

direct product

C₂×#30234

Subgroups

	subgroup	G/H
center, socle, prime radical	C₂⁴	C₂²×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ C₂×C₄³ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1586 conjugacy classes of subgroups; only the 578 normal subgroups are shown.

normal subgroups only

This group has 1586 conjugacy classes of subgroups — too many to draw. Only its 578 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 — 5 generators on 42 points:

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

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, 6343536);