Finite Group Databasebeta

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Cayley diagram

C2×#28652 #6464053

order512
nilpotency class3
isoclinismisoclinic class of #24274
automorphism?
homologyAbelianization (H1) = C25 · Schur multiplier (H2) = C28

Aliases

direct productC2×#28652

Subgroups

subgroupG/H
center, socle, prime radicalC24C22×D4
commutator, Frattini, cosocleC22×C4C25
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C22×C4×Q8 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

2012 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)(7,8,10,11)(9,13,14,12)(15,16,18,21)(17,23,25,20)(19,22,27,29)(24,28,26,30);(8,11)(12,13)(15,17,19,26,18,25,27,24)(16,20,22,28,21,23,29,30);(3,4,5,6)(16,21)(17,24)(19,27)(20,28)(23,30)(25,26);(7,9,10,14)(8,12,11,13)(16,21)(17,25)(22,29)(24,26)

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6371249);