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Cayley diagram

C4×#3331 #6384124

order512
nilpotency class3
isoclinismisoclinic class of #3264
automorphism?
homologyAbelianization (H1) = C24×C4 · Schur multiplier (H2) = C28

Aliases

direct productC4×#3331

Subgroups

subgroupG/H
centerC22×C4C22×D4
commutatorC2×C4C24×C4
Frattini, cosocleC22×C4C25
socleC23C23×D4
prime radicalC2×#547C22
LCSG ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C4 ◁ C42×D4 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

2200 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 28
transitive degreeμt(G) = 256

A minimal faithful permutation representation — 5 generators on 28 points:

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

Conjugacy classes & character table

116 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#72282481
maximal normal subgroup of#72282481

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6291320);