Finite Group Databasebeta

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Cayley diagram

C22×#2634 #7627756

order512
nilpotency class2
isoclinismisoclinic class of #2613
automorphism?
homologyAbelianization (H1) = C26 · Schur multiplier (H2) = C213

Aliases

direct productC22×#2634

Subgroups

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

8402 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 32
transitive degreeμt(G) = 512

A minimal faithful permutation representation — 6 generators on 32 points:

(1,2);(3,4);(10,13)(14,16)(17,18,22,20)(19,25,27,23)(21,24,29,28)(26,31,32,30);(6,8)(10,13)(14,16)(17,19,22,27)(18,23,20,25)(21,26,29,32)(24,30,28,31);(5,6)(7,8)(9,10)(11,14)(12,13)(15,16)(17,20,22,18)(19,23,27,25)(21,24,29,28)(26,31,32,30);(9,11,12,15)(10,14,13,16)(17,21)(18,24)(19,26)(20,28)(22,29)(23,30)(25,31)(27,32)

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, 7534952);