Finite Group Databasebeta

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Cayley diagram

C2×#47637 #7701538

order512
nilpotency class2
isoclinismisoclinic class of #46350
automorphism?
homologyAbelianization (H1) = C26 · Schur multiplier (H2) = C212

Aliases

direct productC2×#47637

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

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofC2×#8657162, C2×#10089117, C2×#7937604
maximal normal subgroup ofC2×#8657162, C2×#10089117, C2×#7937604

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7608734);