C₂×#33288 #6630680

order	512
nilpotency class	3
isoclinism	isoclinic class of #32935
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁶×C₄

Aliases

direct product

C₂×#33288

Subgroups

	subgroup	G/H
center, socle	C₂⁴	C₂²×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	C₂³×D₄	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ C₂×C₄³ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1864 conjugacy classes of subgroups; only the 610 normal subgroups are shown.

normal subgroups only

This group has 1864 conjugacy classes of subgroups — too many to draw. Only its 610 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)(5,9)(6,10)(7,8)(11,12)(13,15)(14,16)(17,18)(19,20)(21,27)(22,30)(23,25)(24,29)(26,31)(28,33)(32,34);(4,8)(9,10)(11,13,14,17)(12,15,16,18)(21,28)(24,32)(27,33)(29,34);(3,5,7,6)(4,9,8,10)(19,21,22,29,23,28,31,34)(20,24,26,27,25,32,30,33);(3,6,7,5)(4,10,8,9)(12,16)(15,18)(20,25)(21,28)(26,30)(29,34)

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6537876);