Finite Group Databasebeta

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Cayley diagram

C2×#37832 #7635145

order512
nilpotency class2
isoclinismisoclinic class of #37644
automorphism?
homologyAbelianization (H1) = C26 · Schur multiplier (H2) = C213

Aliases

direct productC2×#37832

Subgroups

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

17564 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 22
transitive degreeμt(G) = 64

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

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

Conjugacy classes & character table

116 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofC2×#7849701, #72232088, #72232096, …
maximal normal subgroup ofC2×#7849701, #72232088, #72232096, …
maximal quotient of#72232088, #72232106, C2×#846041, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7542341);