Finite Group Databasebeta

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Cayley diagram

C2×#25948 #6447750

order512
nilpotency class3
isoclinismisoclinic class of #24404
automorphism?
homologyAbelianization (H1) = C25 · Schur multiplier (H2) = C28×C4

Aliases

direct productC2×#25948

Subgroups

subgroupG/H
center, socleC23C2×#120
commutator, Frattini, cosocleC24C25
prime radicalC23×D4C23
LCSG ▷ C24 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C24×C4 ◁ G
LECSG ▷ C24 ▷ C2 ▷ C1

3456 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 26
transitive degreeμt(G) = 128

A minimal faithful permutation representation — 5 generators on 26 points:

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

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#72268290, #72268294
maximal normal subgroup of#72268290, #72268294

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6354946);