Finite Group Databasebeta

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Cayley diagram

C2×#56915 #8100010

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

Aliases

direct productC2×#56915

Subgroups

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

5618 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)(5,7)(6,9)(8,10)(11,12,15,17)(13,18,14,16)(20,23)(24,26);(4,5)(6,10)(11,13)(12,16)(14,15)(17,18)(21,25)(24,26);(4,5)(8,9)(13,14)(16,18)(19,20,22,23)(21,24,25,26);(3,5,7,4)(6,9,10,8)(11,14,15,13)(12,16,17,18)(19,20)(21,24)(22,23)(25,26);(3,6,7,10)(4,8,5,9)(11,12)(13,18)(14,16)(15,17)(19,21,22,25)(20,24,23,26)

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8007206);