Finite Group Databasebeta

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Cayley diagram

C4×#1773 #353243

order512
nilpotency class3
isoclinismisoclinic class of AGL(1,ℤ/8)
automorphism?
homologyAbelianization (H1) = C2×C43 · Schur multiplier (H2) = C22×C43

Aliases

direct productC4×#1773

Subgroups

subgroupG/H
centerC2×C42C2×D4
commutatorC4C2×C43
Frattini, cosocleC23×C4C24
socleC23C2×C4×D4
prime radicalC24C2×#41
LCSG ▷ C4 ▷ C2 ▷ C1
UCSC1 ◁ C2×C42 ◁ C22×C4×C8 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

1430 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);(10,14,16,19)(11,17)(15,22,23,21)(18,24);(5,6,7,8)(9,10)(11,15)(12,19)(13,16)(14,20)(17,23)(18,22)(21,24);(9,11,12,18,13,17,20,24)(10,15,19,22,16,23,14,21)

Conjugacy classes & character table

176 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#72212995, C4×#12647, C4×#20512, …
maximal normal subgroup of#72212995, C4×#12647, C4×#20512, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 260439);