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Cayley diagram

C4×#1740 #353148

order512
nilpotency class3
isoclinismisoclinic class of #99
automorphism?
homologyAbelianization (H1) = C2×C43 · Schur multiplier (H2) = C24×C43

Aliases

direct productC4×#1740

Subgroups

subgroupG/H
centerC2×C42C22⋊C4
commutatorC22C2×C43
Frattini, prime radical, cosocleC25C24
socleC23C22×C22⋊C4
LCSG ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C2×C42 ◁ C23×C42 ◁ G
LECSG ▷ C25 ▷ C2 ▷ C1

2918 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);(9,10)(11,17,19,14)(12,15)(13,16)(18,23,24,21)(20,22);(5,6,7,8)(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 ofD4×#1740, C4×#23443, #72310652, …
maximal normal subgroup ofD4×#1740, C4×#23443, #72310652, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 260344);