Finite Group Databasebeta

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Cayley diagram

C2×#53959 #8102462

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

Aliases

direct productC2×#53959

Subgroups

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

5908 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);(3,4,6,7)(5,8,9,10)(12,15)(13,17)(20,23)(21,25);(5,9)(8,10)(11,12,14,15)(13,18,17,16)(20,23)(24,26);(4,7)(8,10)(11,12,14,15)(13,16,17,18)(19,20)(21,24)(22,23)(25,26);(3,4)(5,8)(6,7)(9,10)(11,12)(13,16)(14,15)(17,18)(19,21)(20,24)(22,25)(23,26);(3,5)(4,8)(6,9)(7,10)(11,13,14,17)(12,16,15,18)

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofC2×#8232470, C2×#8525425, C2×#9786712
maximal normal subgroup ofC2×#8232470, C2×#8525425, C2×#9786712
maximal quotient ofC2×#2807749

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8009658);