Finite Group Databasebeta

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Cayley diagram

C4×#1506 #182616

order512
nilpotency class3
isoclinismisoclinic class of #1504
automorphism?
homologyAbelianization (H1) = C22×C42 · Schur multiplier (H2) = C23×C42

Aliases

direct productC4×#1506

Subgroups

subgroupG/H
centerC22×C4C2×C22⋊C4
commutatorC2×C4C22×C42
Frattini, cosocleC23×C4C24
socleC23C22×C22⋊C4
prime radicalC24C22×D4
LCSG ▷ C2×C4 ▷ C22 ▷ C1
UCSC1 ◁ C22×C4 ◁ C2×C43 ◁ G
LECSG ▷ C23×C4 ▷ C22 ▷ C1

1346 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 4 generators on 28 points:

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

Conjugacy classes & character table

128 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#72239862, #72239898
maximal normal subgroup of#72239862, #72239898

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 89812);