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Cayley diagram

C2×C8×Q32 #483494

order512
nilpotency class4
isoclinismisoclinic class of D16
automorphism?
homologyAbelianization (H1) = C23×C8 · Schur multiplier (H2) = C25

Aliases

direct productC2×C8×Q32

Subgroups

subgroupG/H
centerC22×C8D8
commutatorC8C23×C8
Frattini, cosocleC4×C8C24
socle, prime radicalC23C4×D8
LCSG ▷ C8 ▷ C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C8 ◁ C2×C4×C8 ◁ C2×C82 ◁ G
LECSG ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1

514 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 42
transitive degreeμt(G) = 512

A minimal faithful permutation representation — 4 generators on 42 points:

(1,2);(3,4,5,6,7,8,9,10);(11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26)(27,42,41,40,39,38,37,36,35,34,33,32,31,30,29,28);(11,27,19,35)(12,28,20,36)(13,29,21,37)(14,30,22,38)(15,31,23,39)(16,32,24,40)(17,33,25,41)(18,34,26,42)

Conjugacy classes & character table

176 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofC2×C8×#508, C2×C8×SD64, C2×#151078
maximal normal subgroup ofC2×C8×#508, C2×C8×SD64, C2×#151078

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 390690);