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Cayley diagram

C4×#1733 #353130

order512
nilpotency class3
isoclinismisoclinic class of AGL(1,ℤ/8)
automorphism?
homologyAbelianization (H1) = C2×C43 · Schur multiplier (H2) = C25×C42

Aliases

direct productC4×#1733

Subgroups

subgroupG/H
centerC2×C42C2×D4
commutatorC4C2×C43
Frattini, cosocleC23×C4C24
socleC23C22×C22⋊C4
prime radicalC25C2×D4
LCSG ▷ C4 ▷ C2 ▷ C1
UCSC1 ◁ C2×C42 ◁ C23×C42 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

2790 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)(11,17)(12,20)(13,16)(14,19)(15,21)(18,23)(22,24);(9,11)(10,14,16,22)(12,19)(13,18)(15,17,20,23)(21,24);(11,18)(14,22)(17,23)(19,24)

Conjugacy classes & character table

176 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofC4×#10994
maximal normal subgroup ofC4×#10994
maximal quotient of#72213013

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 260326);