Finite Group Databasebeta

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Cayley diagram

C2×#35719 #7629311

order512
nilpotency class2
isoclinismisoclinic class of #35680
automorphism?
homologyAbelianization (H1) = C26 · Schur multiplier (H2) = C212

Aliases

direct productC2×#35719

Subgroups

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

7252 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);(11,12)(13,14)(15,17)(16,18)(19,20)(21,22)(23,25)(24,26);(3,4)(5,8)(6,9)(7,10)(11,13)(12,15)(14,18)(16,17)(19,20)(21,22)(23,25)(24,26);(3,5)(4,7)(6,10)(8,9)(11,14,16,15)(12,17,18,13)(19,21)(20,24)(22,25)(23,26);(3,5)(4,8)(6,10)(7,9)(11,13)(12,14)(15,18)(16,17)(19,22,23,24)(20,21,25,26);(3,4)(5,7)(6,9)(8,10)(11,15,16,14)(12,17,18,13)(19,21)(20,22)(23,26)(24,25)

Conjugacy classes & character table

128 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofC2×#7629359, C2×#7851840
maximal normal subgroup ofC2×#7629359, C2×#7851840

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7536507);