Finite Group Databasebeta

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Cayley diagram

C8×#410 #182061

order512
nilpotency class3
isoclinismisoclinic class of #99
automorphism?
homologyAbelianization (H1) = C22×C4×C8 · Schur multiplier (H2) = C24×C42

Aliases

direct productC8×#410

Subgroups

subgroupG/H
centerC4×C8C22⋊C4
commutatorC22C22×C4×C8
Frattini, cosocleC23×C4C24
socleC22C2×C4×C22⋊C4
prime radicalC22×#41C2×C4
LCSG ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C4×C8 ◁ C22×C4×C8 ◁ G
LECSG ▷ C23×C4 ▷ C22 ▷ C1

1272 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 24
transitive degreeμt(G) = 128

A minimal faithful permutation representation — 4 generators on 24 points:

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

Conjugacy classes & character table

176 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#72342986, #72342995, D8×#410, …
maximal normal subgroup of#72342986, #72342995, D8×#410, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 89257);