Finite Group Databasebeta

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Cayley diagram

C4×#3323 #6384118

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

Aliases

direct productC4×#3323

Subgroups

subgroupG/H
centerC22×C4C22×D4
commutatorC2×C4C24×C4
Frattini, cosocleC22×C4C25
socleC23C23×D4
prime radicalC2×#546C22
LCSG ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C4 ◁ C42×D4 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

2278 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 28
transitive degreeμt(G) = 128

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

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

Conjugacy classes & character table

116 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#72282899
maximal normal subgroup of#72282899

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6291314);