Finite Group Databasebeta

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Cayley diagram

C22×#2767 #7633539

order512
nilpotency class2
isoclinismisoclinic class of #2668
automorphism?
homologyAbelianization (H1) = C26 · Schur multiplier (H2) = C212

Aliases

direct productC22×#2767

Subgroups

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

6002 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);(5,6,9,8)(7,11,12,10)(15,18)(16,20)(22,25)(23,27);(6,8)(7,12)(13,14,17,19)(15,16,18,20)(21,22,24,25)(23,28,27,26);(5,7,9,12)(6,10,8,11)(13,15,17,18)(14,16,19,20);(5,8,9,6)(7,11,12,10)(13,16)(14,18)(15,19)(17,20)(21,23,24,27)(22,26,25,28)

Conjugacy classes & character table

128 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7540735);