Finite Group Databasebeta

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Cayley diagram

C2×#51052 #7707191

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

Aliases

direct productC2×#51052

Subgroups

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

6486 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)(5,9)(6,7)(8,10)(12,14)(16,17)(21,25)(23,26);(3,5)(4,8)(6,9)(7,10)(11,12,15,14)(13,17,18,16)(19,20,22,24)(21,23,25,26);(5,10)(8,9)(19,21)(20,23)(22,25)(24,26);(3,5)(4,9)(6,8)(7,10)(11,13,15,18)(12,16,14,17);(3,6,7,4)(5,8,10,9)(11,14,15,12)(13,17,18,16)

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofC2×#7837040, C2×#7837055, C2×#7707199
maximal normal subgroup ofC2×#7837040, C2×#7837055, C2×#7707199

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7614387);