C₂²×#2863 #7633338

order	512
nilpotency class	2
isoclinism	isoclinic class of #2846
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹²

Aliases

direct product

C₂²×#2863

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

7410 conjugacy classes of subgroups; only the 3906 normal subgroups are shown.

normal subgroups only

This group has 7410 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);(5,6,10,9)(7,12,8,11)(13,14,17,20)(15,18,19,16)(23,27)(25,28);(5,7,10,8)(6,11,9,12)(13,15,17,19)(14,18,20,16)(21,22,24,26)(23,25,27,28);(5,8,10,7)(6,12,9,11)(13,16,17,18)(14,19,20,15);(5,9,10,6)(7,12,8,11)(13,14)(15,18)(16,19)(17,20)(21,23)(22,25)(24,27)(26,28)

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, 7540534);