Finite Group Databasebeta

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Cayley diagram

C2×#15854 #609583

order512
nilpotency class2
isoclinismisoclinic class of #15257
automorphism?
homologyAbelianization (H1) = C25 · Schur multiplier (H2) = C26×C42

Aliases

direct productC2×#15854

Subgroups

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

4106 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)(8,11)(9,13)(15,16,19,17)(18,21,22,20)(25,29)(28,30);(8,11)(12,14)(16,17)(18,22)(23,24,27,26)(25,28,29,30);(7,8,10,11)(9,12,13,14)(15,17,19,16)(18,21,22,20)(23,25)(24,28)(26,30)(27,29);(3,4)(5,6)(7,9,10,13)(8,12,11,14)(15,18)(16,20)(17,21)(19,22)(23,26,27,24)(25,30,29,28)

Conjugacy classes & character table

116 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 516779);