Finite Group Databasebeta

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Cayley diagram

C2×#30919 #7502268

order512
nilpotency class3
isoclinismisoclinic class of #30396
automorphism?
homologyAbelianization (H1) = C25 · Schur multiplier (H2) = C27×C4

Aliases

direct productC2×#30919

Subgroups

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

2936 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);(7,8,10,11)(9,13,14,12)(15,16)(17,21)(18,22)(19,20);(3,4)(5,6)(8,11)(9,14)(15,17,18,19)(16,20,22,21);(4,6)(16,20)(17,19)(21,22);(7,9,10,14)(8,12,11,13)(16,21)(17,19)(20,22)

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofC2×#7515180, C2×#7518380, C2×#7502680
maximal normal subgroup ofC2×#7515180, C2×#7518380, C2×#7502680

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7409464);