Finite Group Databasebeta

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Cayley diagram

C2×#35419 #7628779

order512
nilpotency class2
isoclinismisoclinic class of #35346
automorphism?
homologyAbelianization (H1) = C26 · Schur multiplier (H2) = C212

Aliases

direct productC2×#35419

Subgroups

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

6786 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);(11,12,15,14)(13,17,18,16)(19,20,24,22)(21,25,26,23);(3,4)(5,9)(6,7)(8,10)(11,13)(12,16)(14,17)(15,18)(19,21)(20,25)(22,23)(24,26);(4,7)(8,9)(13,18)(16,17)(20,22)(21,26);(3,4)(5,8)(6,7)(9,10)(11,13)(12,17)(14,16)(15,18)(19,22,24,20)(21,23,26,25);(3,5)(4,8)(6,10)(7,9)(11,14,15,12)(13,16,18,17)(19,23)(20,21)(22,26)(24,25)

Conjugacy classes & character table

128 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofC2×#7628788
maximal normal subgroup ofC2×#7628788

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7535975);