Finite Group Databasebeta

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Cayley diagram

C22×#2507 #7627935

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

Aliases

direct productC22×#2507

Subgroups

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

7002 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)(7,11)(8,9)(10,12)(14,16)(15,19)(21,22,24,26)(23,27,28,25);(6,9)(10,11)(13,14,17,16)(15,20,19,18)(21,22)(23,25)(24,26)(27,28);(5,6)(7,10)(8,9)(11,12)(13,15,17,19)(14,18,16,20);(5,7,8,12)(6,10,9,11)(13,16,17,14)(15,20,19,18)(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, 7535131);