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Cayley diagram

C2×#17072 #612281

order512
nilpotency class2
isoclinismisoclinic class of #16249
automorphism?
homologyAbelianization (H1) = C25 · Schur multiplier (H2) = C28×C4

Aliases

direct productC2×#17072

Subgroups

subgroupG/H
center, socle, prime radicalC25C24
commutator, Frattini, cosocleC24C25
LCSG ▷ C24 ▷ C1
UCSC1 ◁ C25 ◁ G
LECSG ▷ C24 ▷ C1

4886 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 22
transitive degreeμt(G) = 512

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

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

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

Relationships

Fitting subgroup ofC2×#17072⋊C3
maximal subgroup of#72267035, C2×#17072⋊C3, #72267457
maximal normal subgroup of#72267035, C2×#17072⋊C3, #72267457

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 519477);