Finite Group Databasebeta

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Cayley diagram

C2×#47924 #7703948

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

Aliases

direct productC2×#47924

Subgroups

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

6348 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);(11,12,16,17)(13,18,15,14)(20,22)(24,25);(3,4)(5,9)(6,10)(7,8)(12,17)(13,15)(19,20,23,22)(21,25,26,24);(4,8)(5,10)(11,13,16,15)(12,18,17,14)(19,20,23,22)(21,24,26,25);(3,5)(4,6)(7,10)(8,9)(11,14,16,18)(12,13,17,15)(19,21,23,26)(20,24,22,25);(3,6)(4,5)(7,9)(8,10)(11,15,16,13)(12,18,17,14)(19,22,23,20)(21,25,26,24)

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, 7611144);