C₂×#32814 #6631479

order	512
nilpotency class	3
isoclinism	isoclinic class of #32712
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁴×C₄³

Aliases

direct product

C₂×#32814

Subgroups

	subgroup	G/H
center, socle	C₂⁴	C₂²×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ C₂×#376 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2382 conjugacy classes of subgroups; only the 538 normal subgroups are shown.

normal subgroups only

This group has 2382 conjugacy classes of subgroups — too many to draw. Only its 538 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) = 34
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 34 points:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6538675);