C₂²×#2864 #7633339

order	512
nilpotency class	2
isoclinism	isoclinic class of #2846
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹¹×C₄

Aliases

direct product

C₂²×#2864

Subgroups

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

7090 conjugacy classes of subgroups; only the 3906 normal subgroups are shown.

normal subgroups only

This group has 7090 conjugacy classes of subgroups — too many to draw. Only its 3906 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) = 28
transitive degree	μ_t(G) = 512

A minimal faithful permutation representation — 6 generators on 28 points:

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

Conjugacy classes & character table

176 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7540535);