Finite Group Databasebeta

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Cayley diagram

C2×#59093 #7666169

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

Aliases

direct productC2×#59093

Subgroups

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

10080 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);(7,8)(9,13)(10,11)(12,14)(16,19)(17,21);(4,6)(8,11)(12,13)(16,19)(17,21);(15,16)(17,20)(18,19)(21,22);(3,4)(5,6)(7,8)(9,12)(10,11)(13,14)(15,17,18,21)(16,20,19,22);(7,9,10,14)(8,12,11,13)

Conjugacy classes & character table

128 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#72261030, C2×#7699554, #72261676, …
maximal normal subgroup of#72261030, C2×#7699554, #72261676, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7573365);