Finite Group Databasebeta

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Cayley diagram

C2×#28952 #6460215

order512
nilpotency class3
isoclinismisoclinic class of #24274
automorphism?
homologyAbelianization (H1) = C25 · Schur multiplier (H2) = C26×C4

Aliases

direct productC2×#28952

Subgroups

subgroupG/H
center, socleC24C22×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC23×D4C23
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C2×C4×C4⋊C4 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

1968 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 26
transitive degreeμt(G) = 256

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

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

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofC2×#6741717, C2×#6460700, C2×#6699654
maximal normal subgroup ofC2×#6741717, C2×#6460700, C2×#6699654

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6367411);