C₂²×#2817 #7633381

order	512
nilpotency class	2
isoclinism	isoclinic class of #2812
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹³

Aliases

direct product

C₂²×#2817

Subgroups

	subgroup	G/H
center, socle	C₂⁵	C₂⁴
commutator, Frattini, cosocle	C₂³	C₂⁶
prime radical	C₂⁶	C₂³
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂⁵ ◁ G
LECS	G ▷ C₂³ ▷ C₁

9258 conjugacy classes of subgroups; only the 3490 normal subgroups are shown.

normal subgroups only

This group has 9258 conjugacy classes of subgroups — too many to draw. Only its 3490 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) = 32
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 6 generators on 32 points:

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

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, 7540577);