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Cayley diagram

C2×#27777 #6457650

order512
nilpotency class3
isoclinismisoclinic class of #24269
automorphism?
homologyAbelianization (H1) = C25 · Schur multiplier (H2) = C27×C4

Aliases

direct productC2×#27777

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

2154 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);(4,5)(7,8)(9,11)(10,12)(13,14)(15,16,19,23,18,22,25,29)(17,24,27,30,26,20,28,21);(3,4,6,5)(7,9,10,13)(8,11,12,14)(15,17,25,28,18,26,19,27)(16,20,29,30,22,24,23,21);(8,12)(11,14)(15,17,18,26)(16,21,22,30)(19,28,25,27)(20,29,24,23);(3,5,6,4)

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, 6364846);