Finite Group Databasebeta

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Cayley diagram

C2×#17080 #612320

order512
nilpotency class2
isoclinismisoclinic class of #16249
automorphism?
homologyAbelianization (H1) = C25 · Schur multiplier (H2) = C26×C42

Aliases

direct productC2×#17080

Subgroups

subgroupG/H
center, socle, prime radicalC25C24
commutator, Frattini, cosocleC24C25
LCSG ▷ C24 ▷ C1
UCSC1 ◁ C25 ◁ G
LECSG ▷ C24 ▷ C1

4342 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 30
transitive degreeμt(G) = 512

A minimal faithful permutation representation — 5 generators on 30 points:

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

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 519516);