Finite Group Databasebeta

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Cayley diagram

C4×#2518 #515004

order512
nilpotency class2
isoclinismisoclinic class of #2403
automorphism?
homologyAbelianization (H1) = C24×C4 · Schur multiplier (H2) = C26×C4

Aliases

direct productC4×#2518

Subgroups

subgroupG/H
centerC23×C4C24
commutatorC23C24×C4
Frattini, socle, cosocleC24C25
prime radicalC23×D4C23
LCSG ▷ C23 ▷ C1
UCSC1 ◁ C23×C4 ◁ G
LECSG ▷ C24 ▷ C1

2848 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 28
transitive degreeμt(G) = 256

A minimal faithful permutation representation — 5 generators on 28 points:

(1,2,3,4);(5,6,8,9)(7,10,11,12)(13,14)(15,18)(16,17)(19,20)(21,22,24,26)(23,25,27,28);(6,9)(10,12)(21,23)(22,25)(24,27)(26,28);(5,7,8,11)(6,10,9,12)(14,17)(18,20)(23,27)(25,28);(6,9)(10,12)(13,15,16,19)(14,18,17,20)

Conjugacy classes & character table

152 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#72304042, #72304102
maximal normal subgroup of#72304042, #72304102

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 422200);