Finite Group Databasebeta

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Cayley diagram

C22×#2514 #7627944

order512
nilpotency class2
isoclinismisoclinic class of #2403
automorphism?
homologyAbelianization (H1) = C26 · Schur multiplier (H2) = C212

Aliases

direct productC22×#2514

Subgroups

subgroupG/H
center, socle, prime radicalC25C24
commutator, Frattini, cosocleC23C26
LCSG ▷ C23 ▷ C1
UCSC1 ◁ C25 ◁ G
LECSG ▷ C23 ▷ C1

6714 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 28
transitive degreeμt(G) = 512

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

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

Conjugacy classes & character table

152 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7535140);