Finite Group Databasebeta

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Cayley diagram

C22×#2605 #7628026

order512
nilpotency class2
isoclinismisoclinic class of #2403
automorphism?
homologyAbelianization (H1) = C26 · Schur multiplier (H2) = C211×C4

Aliases

direct productC22×#2605

Subgroups

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

7338 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);(13,14,17,16)(15,19,20,18)(21,22)(23,27)(24,26)(25,28);(6,9)(10,12)(15,20)(18,19)(21,22,24,26)(23,25,28,27);(5,6)(7,10)(8,9)(11,12)(13,15,17,20)(14,18,16,19)(21,22)(23,25)(24,26)(27,28);(5,7,8,11)(6,10,9,12)(13,16,17,14)(15,19,20,18)(21,23,24,28)(22,25,26,27)

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, 7535222);