Finite Group Databasebeta

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Cayley diagram

C2×#18165 #6346419

order512
nilpotency class3
isoclinismisoclinic class of #18162
automorphism?
homologyAbelianization (H1) = C25 · Schur multiplier (H2) = C210

Aliases

direct productC2×#18165

Subgroups

subgroupG/H
center, socleC23C2×#120
commutator, Frattini, cosocleC22×C4C25
prime radicalC2×#3492C2
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C24×C4 ◁ G
LECSG ▷ C22×C4 ▷ C22 ▷ C1

4090 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)(11,12,16,19)(13,20,22,25)(14,18,15,23)(17,24,26,21);(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)(4,7)(6,9)(8,10)(11,14,16,15)(12,18,19,23)(13,21,22,24)(17,25,26,20);(13,22)(17,26)(20,25)(21,24)

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofC2×#6346595, C2×#6346600, C2×#6346423
maximal normal subgroup ofC2×#6346595, C2×#6346600, C2×#6346423

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6253615);