Finite Group Databasebeta

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Cayley diagram

C2×#18114 #6346353

order512
nilpotency class3
isoclinismisoclinic class of #1595
automorphism?
homologyAbelianization (H1) = C25 · Schur multiplier (H2) = C210

Aliases

direct productC2×#18114

Subgroups

subgroupG/H
centerC22×C4#120
commutator, Frattini, cosocleC22×C4C25
socleC23C2×#120
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C22×C4 ◁ C24×C4 ◁ G
LECSG ▷ C22×C4 ▷ C22 ▷ C1

4176 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);(7,10)(11,12)(13,20)(14,18)(15,23)(16,19)(17,21)(22,25)(24,26);(3,4)(5,7)(6,8)(9,10)(11,13)(12,17)(14,21)(15,24)(16,22)(18,25)(19,26)(20,23);(3,5,6,9)(4,7,8,10);(11,14,16,15)(12,18,19,23)(13,21,22,24)(17,25,26,20)

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofC2×#6346423, C2×#6346468, C2×#6346389
maximal normal subgroup ofC2×#6346423, C2×#6346468, C2×#6346389

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6253549);