Finite Group Databasebeta

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Cayley diagram

C2×#50828 #7706283

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

Aliases

direct productC2×#50828

Subgroups

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

6150 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);(3,4)(5,9)(6,7)(8,10)(21,25)(24,26);(4,7)(8,9)(11,12)(13,15)(14,17)(16,18)(19,20,23,22)(21,24,25,26);(12,13)(16,17)(19,21)(20,24)(22,26)(23,25);(3,4)(5,8)(6,7)(9,10)(11,13,15,12)(14,17,18,16)(19,20,23,22)(21,24,25,26);(3,5)(4,8)(6,10)(7,9)(11,14)(12,16)(13,17)(15,18)(19,22,23,20)(21,26,25,24)

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofC2×#10320757
maximal normal subgroup ofC2×#10320757

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7613479);