Finite Group Databasebeta

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Cayley diagram

C2×#40528 #7639332

order512
nilpotency class2
isoclinismisoclinic class of #40484
automorphism?
homologyAbelianization (H1) = C26 · Schur multiplier (H2) = C213

Aliases

direct productC2×#40528

Subgroups

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

8264 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 22
transitive degreeμt(G) = 128

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

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

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofC2×#7659538, C2×#7952445, C2×#7959865
maximal normal subgroup ofC2×#7659538, C2×#7952445, C2×#7959865
maximal quotient ofC2×#1506963, C2×#1001003

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7546528);