Finite Group Databasebeta

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Cayley diagram

C2×#24262 #6442722

order512
nilpotency class3
isoclinismisoclinic class of #23903
automorphism?
homologyAbelianization (H1) = C25 · Schur multiplier (H2) = C29×C4

Aliases

direct productC2×#24262

Subgroups

subgroupG/H
center, socleC23C2×#120
commutator, Frattini, cosocleC24C25
LCSG ▷ C24 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C24×C4 ◁ G
LECSG ▷ C24 ▷ C2 ▷ C1

5642 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 5 generators on 22 points:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#72240328, #72240329, C2×#511938, …
maximal normal subgroup of#72240328, #72240329, C2×#511938, …
maximal quotient of#72261512, C2×#393934, #72316253

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6349918);