Finite Group Databasebeta

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Cayley diagram

C2×#23462 #6431826

order512
nilpotency class3
isoclinismisoclinic class of #23400
automorphism?
homologyAbelianization (H1) = C25 · Schur multiplier (H2) = C210

Aliases

direct productC2×#23462

Subgroups

subgroupG/H
center, socleC23C2×#120
commutator, Frattini, cosocleC22×C4C25
prime radicalC23×#41C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C24×C4 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

4070 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);(3,4,6,8)(5,9,10,7)(11,12)(13,19)(14,22)(15,18)(16,21)(17,23)(20,26)(24,25);(3,5,6,10)(4,7,8,9)(14,23)(16,25)(17,22)(21,24);(11,13)(12,16)(14,24)(15,20)(17,26)(18,25)(19,22)(21,23);(13,20)(16,25)(19,26)(21,24)

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#72262065
maximal normal subgroup of#72262065

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6339022);