Finite Group Databasebeta

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Cayley diagram

C22×#2590 #7628005

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

Aliases

direct productC22×#2590

Subgroups

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

6698 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);(6,9)(10,12)(13,14,17,18)(15,20,19,16)(23,27)(26,28);(6,9)(10,12)(14,18)(15,19)(21,22,25,24)(23,26,27,28);(5,6,8,9)(7,10,11,12)(13,15,17,19)(14,16,18,20)(21,23)(22,26)(24,28)(25,27);(5,7,8,11)(6,10,9,12)(13,16)(14,19)(15,18)(17,20)(21,24,25,22)(23,28,27,26)

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, 7535201);