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Cayley diagram

C2×#13399 #499052

order512
nilpotency class4
isoclinismisoclinic class of #13398
automorphism?
homologyAbelianization (H1) = C24 · Schur multiplier (H2) = C26

Aliases

direct productC2×#13399

Subgroups

subgroupG/H
center, socleC23#481
commutator, Frattini, cosocleC22×C8C24
prime radicalC25SD16
LCSG ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C23×C4 ◁ C23×M16 ◁ G
LECSG ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1

1100 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 38
transitive degreeμt(G) = 128

A minimal faithful permutation representation — 4 generators on 38 points:

(1,2);(7,8,12,16)(9,17,18,35)(10,21,22,38)(11,26,24,15)(13,28,29,19)(14,31,32,25)(20,33,27,36)(23,30,37,34);(3,4)(5,6)(10,22)(14,32)(17,35)(19,28)(21,38)(23,37)(25,31)(33,36);(4,6)(7,9,11,27,12,18,24,20)(8,13,15,34,16,29,26,30)(10,23,25,28,22,37,31,19)(14,33,21,35,32,36,38,17)

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#72282164, #72282220, C2×#499265
maximal normal subgroup of#72282164, #72282220, C2×#499265

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 406248);