C₂×#29295 #6468381

order	512
nilpotency class	3
isoclinism	isoclinic class of #23924
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷

Aliases

direct product

C₂×#29295

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×#376 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1718 conjugacy classes of subgroups; only the 698 normal subgroups are shown.

normal subgroups only

This group has 1718 conjugacy classes of subgroups — too many to draw. Only its 698 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) = 512

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

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

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, 6375577);