Finite Group Databasebeta

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Cayley diagram

C2×#25836 #6445725

order512
nilpotency class3
isoclinismisoclinic class of #23848
automorphism?
homologyAbelianization (H1) = C25 · Schur multiplier (H2) = C29

Aliases

direct productC2×#25836

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

3162 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)(7,8,11,14)(9,13,10,12)(15,16)(17,21)(18,26)(19,28)(20,23)(22,27)(24,30)(25,29);(7,9,11,10)(8,12,14,13)(15,17,20,25)(16,21,23,29)(18,27,24,19)(22,30,28,26);(3,4,5,6)(15,18,19,17,20,24,27,25)(16,21,22,26,23,29,28,30);(7,10,11,9)(8,13,14,12)

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, 6352921);