Finite Group Databasebeta

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Cayley diagram

C2×#50313 #7705924

order512
nilpotency class2
isoclinismisoclinic class of #47826
automorphism?
homologyAbelianization (H1) = C26 · Schur multiplier (H2) = C212

Aliases

direct productC2×#50313

Subgroups

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

6674 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 26
transitive degreeμt(G) = 128

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

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

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofC2×#10442541
maximal normal subgroup ofC2×#10442541

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7613120);