Finite Group Databasebeta

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Cayley diagram

C2×#52981 #8092820

order512
nilpotency class2
isoclinismisoclinic class of #52308
automorphism?
homologyAbelianization (H1) = C26 · Schur multiplier (H2) = C212

Aliases

direct productC2×#52981

Subgroups

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

7148 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);(5,9)(8,10)(12,14)(13,17)(19,20)(21,22)(23,24)(25,26);(3,4)(5,8)(6,7)(9,10)(11,12,15,14)(13,18,17,16)(20,24)(22,25);(4,7)(8,10)(13,17)(16,18)(19,21)(20,25)(22,24)(23,26);(3,4)(5,8)(6,7)(9,10)(11,13,15,17)(12,16,14,18)(19,22,23,25)(20,21,24,26);(3,5,6,9)(4,8,7,10)(11,14,15,12)(13,18,17,16)

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofC2×#9266284
maximal normal subgroup ofC2×#9266284

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8000016);