Finite Group Databasebeta

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Cayley diagram

C2×#25030 #6444043

order512
nilpotency class3
isoclinismisoclinic class of #24264
automorphism?
homologyAbelianization (H1) = C25 · Schur multiplier (H2) = C28

Aliases

direct productC2×#25030

Subgroups

subgroupG/H
center, socleC24C22×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC22×#121C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C2×C4×C22⋊C4 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

2586 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 22
transitive degreeμt(G) = 256

A minimal faithful permutation representation — 5 generators on 22 points:

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

Conjugacy classes & character table

116 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofC2×#6445075, C2×#6577769, C2×#6574826
maximal normal subgroup ofC2×#6445075, C2×#6577769, C2×#6574826

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6351239);