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Cayley diagram

C2×#25579 #6449041

order512
nilpotency class3
isoclinismisoclinic class of #23860
automorphism?
homologyAbelianization (H1) = C25 · Schur multiplier (H2) = C28×C4

Aliases

direct productC2×#25579

Subgroups

subgroupG/H
center, socleC24C22×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC2×#2540C2
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C22×C4×D4 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

3906 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 5 generators on 30 points:

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

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#72282839
maximal normal subgroup of#72282839

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6356237);