Finite Group Databasebeta

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Cayley diagram

C2×#55244 #8102084

order512
nilpotency class2
isoclinismisoclinic class of #53390
automorphism?
homologyAbelianization (H1) = C26 · Schur multiplier (H2) = C212

Aliases

direct productC2×#55244

Subgroups

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

6230 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);(3,4,6,7)(5,8,9,10)(11,12)(13,17)(14,16)(15,18);(5,9)(8,10)(11,12,14,16)(13,15,18,17)(19,20,23,24)(21,22,25,26);(4,7)(8,10)(20,24)(21,25);(3,4)(5,8)(6,7)(9,10)(11,12)(13,15)(14,16)(17,18)(19,21)(20,22)(23,25)(24,26);(3,5,6,9)(4,8,7,10)(11,13,14,18)(12,15,16,17)(19,22)(20,21)(23,26)(24,25)

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofC2×#10534354
maximal normal subgroup ofC2×#10534354

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8009280);