Finite Group Databasebeta

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Cayley diagram

C2×#10713 #353285

order512
nilpotency class2
isoclinismisoclinic class of #392
automorphism?
homologyAbelianization (H1) = C22×C42 · Schur multiplier (H2) = C29

Aliases

direct productC2×#10713

Subgroups

subgroupG/H
centerC24×C4C23
commutatorC23C22×C42
Frattini, cosocleC23×C4C24
socleC25C22×C4
prime radicalC26C2×C4
LCSG ▷ C23 ▷ C1
UCSC1 ◁ C24×C4 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

5122 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 30
transitive degreeμt(G) = 256

A minimal faithful permutation representation — 4 generators on 30 points:

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

Conjugacy classes & character table

176 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofC2×#354295, C2×#106079
maximal normal subgroup ofC2×#354295, C2×#106079

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 260481);