Finite Group Databasebeta

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Cayley diagram

C2×#25324 #6447069

order512
nilpotency class3
isoclinismisoclinic class of #23848
automorphism?
homologyAbelianization (H1) = C25 · Schur multiplier (H2) = C28

Aliases

direct productC2×#25324

Subgroups

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

2326 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);(3,4,6,5)(7,8,10,11)(9,13,14,12)(15,16,18,21)(17,23,25,20)(19,22,26,29)(24,30,27,28);(3,5,6,4)(8,11)(12,13)(17,24)(19,26)(20,28)(22,29)(23,30)(25,27);(4,5)(15,17,18,25)(16,20,21,23)(19,27,26,24)(22,30,29,28);(7,9,10,14)(8,12,11,13)

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, 6354265);