Finite Group Databasebeta

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Cayley diagram

C2×#53701 #8101940

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

Aliases

direct productC2×#53701

Subgroups

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

5834 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)(5,6)(7,8)(9,10)(20,23)(21,25);(4,8)(5,9)(11,12)(13,17)(14,15)(16,18)(20,23)(24,26);(12,15)(16,17)(19,20,22,23)(21,24,25,26);(3,5,7,9)(4,6,8,10)(11,12)(13,16)(14,15)(17,18)(19,21,22,25)(20,24,23,26);(3,6)(4,9)(5,8)(7,10)(11,13)(12,16)(14,18)(15,17)

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, 8009136);