Finite Group Databasebeta

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Cayley diagram

C2×#24231 #6442599

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

Aliases

direct productC2×#24231

Subgroups

subgroupG/H
center, socleC23C2×#120
commutator, Frattini, cosocleC22×C4C25
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C22×C42 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

4678 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 22
transitive degreeμt(G) = 128

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

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

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofC2×#6442605, C2×#6453883, C2×#6453870
maximal normal subgroup ofC2×#6442605, C2×#6453883, C2×#6453870
maximal quotient ofC2×#261360, #72309294

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6349795);