Finite Group Databasebeta

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Cayley diagram

C2×#30777 #7501856

order512
nilpotency class3
isoclinismisoclinic class of #30396
automorphism?
homologyAbelianization (H1) = C25 · Schur multiplier (H2) = C28

Aliases

direct productC2×#30777

Subgroups

subgroupG/H
center, socleC24C22×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC25C2×D4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C2×C4×C22⋊C4 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

2254 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)(7,8,11,14)(9,10,13,12);(7,9,11,13)(8,12,14,10)(15,16,18,21)(17,19,22,20);(4,6)(7,10,11,12)(8,13,14,9)(16,19)(17,22)(20,21);(4,6)(8,14)(9,13)

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofC2×#7502928, C2×#7523178, C2×#7517084
maximal normal subgroup ofC2×#7502928, C2×#7523178, C2×#7517084

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7409052);