Finite Group Databasebeta

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Cayley diagram

C2×#34232 #7625503

order512
nilpotency class2
isoclinismisoclinic class of #3484
automorphism?
homologyAbelianization (H1) = C25×C4 · Schur multiplier (H2) = C213

Aliases

direct productC2×#34232

Subgroups

subgroupG/H
center, socleC24C25
commutatorC22C25×C4
Frattini, cosocleC23C26
prime radicalC22×#532C2
LCSG ▷ C22 ▷ C1
UCSC1 ◁ C24 ◁ G
LECSG ▷ C23 ▷ C1

9446 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);(3,4,6,5)(7,8,12,10)(9,14,11,13)(16,19)(20,22);(3,5,6,4)(8,10)(13,14);(7,9,12,11)(8,13,10,14)(15,16)(17,20)(18,19)(21,22);(7,10,12,8)(9,14,11,13);(7,11,12,9)(8,14,10,13)(15,17,18,21)(16,20,19,22)

Conjugacy classes & character table

176 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofC2×#7677390, C2×#7763344, C2×#7627268
maximal normal subgroup ofC2×#7677390, C2×#7763344, C2×#7627268

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7532699);