Finite Group Databasebeta

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Cayley diagram

C2×#34257 #7625455

order512
nilpotency class2
isoclinismisoclinic class of #3484
automorphism?
homologyAbelianization (H1) = C25×C4 · Schur multiplier (H2) = C215

Aliases

direct productC2×#34257

Subgroups

subgroupG/H
center, socleC24C25
commutatorC22C25×C4
Frattini, cosocleC23C26
prime radicalC23×2−1+4C2
LCSG ▷ C22 ▷ C1
UCSC1 ◁ C24 ◁ G
LECSG ▷ C23 ▷ C1

23058 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 26
transitive degreeμt(G) = 256

A minimal faithful permutation representation — 6 generators on 26 points:

(1,2);(4,6)(7,8,10,9)(12,13)(15,23)(17,19)(21,26);(11,12,16,13)(14,19,22,17)(15,18,23,20)(21,25,26,24);(3,4,5,6)(7,9,10,8)(11,13,16,12)(14,19,22,17)(15,20,23,18)(21,25,26,24);(11,14,16,22)(12,17,13,19)(15,21,23,26)(18,24,20,25);(11,15)(12,18)(13,20)(14,21)(16,23)(17,24)(19,25)(22,26)

Conjugacy classes & character table

176 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofC2×#7625468, #72302839, #72302843
maximal normal subgroup ofC2×#7625468, #72302839, #72302843

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7532651);