Finite Group Databasebeta

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Cayley diagram

C2×#59792 #7666776

order512
nilpotency class2
isoclinismisoclinic class of #58990
automorphism?
homologyAbelianization (H1) = C26 · Schur multiplier (H2) = C213

Aliases

direct productC2×#59792

Subgroups

subgroupG/H
center, socleC24C25
commutator, Frattini, cosocleC23C26
LCSG ▷ C23 ▷ C1
UCSC1 ◁ C24 ◁ G
LECSG ▷ C23 ▷ C1

7900 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 22
transitive degreeμt(G) = 256

A minimal faithful permutation representation — 6 generators on 22 points:

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

Conjugacy classes & character table

128 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofC2×#7666815, C2×#7698170, C2×#7698179
maximal normal subgroup ofC2×#7666815, C2×#7698170, C2×#7698179

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7573972);