Finite Group Databasebeta

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Cayley diagram

C2×#23956 #6451760

order512
nilpotency class3
isoclinismisoclinic class of #23876
automorphism?
homologyAbelianization (H1) = C25 · Schur multiplier (H2) = C210

Aliases

direct productC2×#23956

Subgroups

subgroupG/H
center, socleC23C2×#120
commutator, Frattini, cosocleC22×C4C25
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C24×C4 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

4280 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 26
transitive degreeμt(G) = 64

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

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

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofC2×#6499844, #72318108, #72318124
maximal normal subgroup ofC2×#6499844, #72318108, #72318124

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6358956);