C₂²×#2843 #7633399

order	512
nilpotency class	2
isoclinism	isoclinic class of #2812
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹²

Aliases

direct product

C₂²×#2843

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

6234 conjugacy classes of subgroups; only the 3490 normal subgroups are shown.

normal subgroups only

This group has 6234 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) = 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)(13,14)(15,19)(16,20)(17,18);(7,11)(9,12)(14,18)(16,19)(21,22,24,26)(23,27,28,25);(5,6)(7,9)(8,10)(11,12)(13,15)(14,19)(16,18)(17,20)(21,23,24,28)(22,25,26,27);(5,7)(6,9)(8,11)(10,12)(13,16,17,19)(14,20,18,15)

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, 7540595);