Finite Group Databasebeta

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Cayley diagram

C2×#30647 #7501395

order512
nilpotency class3
isoclinismisoclinic class of #30412
automorphism?
homologyAbelianization (H1) = C25 · Schur multiplier (H2) = C28

Aliases

direct productC2×#30647

Subgroups

subgroupG/H
center, socleC24C22×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC2×#2428C2
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C2×C4×C22⋊C4 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

2660 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);(3,4)(5,6)(7,8,10,12)(9,11,13,14)(16,19)(20,22);(7,8)(9,11)(10,12)(13,14)(16,20)(18,21)(19,22);(4,6)(7,9,10,13)(8,11,12,14)(15,16,17,19)(18,22,21,20);(4,6)(8,12)(11,14)

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofC2×#7502663, C2×#7521311
maximal normal subgroup ofC2×#7502663, C2×#7521311

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7408591);