Finite Group Databasebeta

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Cayley diagram

C2×#30835 #7501964

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

Aliases

direct productC2×#30835

Subgroups

subgroupG/H
center, socleC24C22×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC22×#127C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C2×C4×C4⋊C4 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

2324 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);(7,8,11,14)(9,10,13,12)(15,16)(17,19)(18,20)(21,22);(3,4)(5,6)(7,9,11,13)(8,12,14,10)(16,19)(20,21);(4,6)(7,10,11,12)(8,13,14,9)(16,20)(18,22)(19,21);(8,14)(9,13)(16,21)(18,22)(19,20)

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofC2×#7518135, C2×#7522927, C2×#7502680
maximal normal subgroup ofC2×#7518135, C2×#7522927, C2×#7502680

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7409160);