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Cayley diagram

C4×#1764 #353209

order512
nilpotency class3
isoclinismisoclinic class of D8
automorphism?
homologyAbelianization (H1) = C2×C43 · Schur multiplier (H2) = C23×C43

Aliases

direct productC4×#1764

Subgroups

subgroupG/H
centerC2×C4×C8D4
commutatorC4C2×C43
Frattini, cosocleC23×C4C24
socleC23C2×C4×D4
prime radicalC22×#41C23
LCSG ▷ C4 ▷ C2 ▷ C1
UCSC1 ◁ C2×C4×C8 ◁ C22×C4×C8 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

1954 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:

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

Conjugacy classes & character table

224 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofC4×#20512
maximal normal subgroup ofC4×#20512

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 260405);