Finite Group Databasebeta

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Cayley diagram

C4×#2039 #359076

order512
nilpotency class3
isoclinismisoclinic class of #2002
automorphism?
homologyAbelianization (H1) = C23×C4 · Schur multiplier (H2) = C26

Aliases

direct productC4×#2039

Subgroups

subgroupG/H
centerC22×C4#120
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
socleC23C2×#120
prime radicalC2×#537C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C4 ◁ C22×C42 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

1684 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 16
transitive degreeμt(G) = 128

A minimal faithful permutation representation — 4 generators on 16 points:

(1,2,3,4);(6,7)(9,10)(11,14)(12,15)(13,16);(5,6)(7,8)(9,10)(11,15)(12,14)(13,16);(5,7,8,6)(10,14)(11,12)(15,16)

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#72282532, C4×#24222
maximal normal subgroup of#72282532, C4×#24222

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 266272);