Finite Group Databasebeta

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Cayley diagram

C2×#47916 #7703701

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

Aliases

direct productC2×#47916

Subgroups

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

5732 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);(11,12)(13,17)(14,16)(15,18)(19,20,23,21)(22,25,26,24);(3,4)(5,9)(6,10)(7,8)(11,12,15,18)(13,16,14,17)(19,21,23,20)(22,25,26,24);(4,8)(5,10)(11,13,15,14)(12,16,18,17)(19,22)(20,24)(21,25)(23,26);(3,5)(4,6)(7,10)(8,9)(11,12)(13,16)(14,17)(15,18);(3,6)(4,5)(7,9)(8,10)(11,14,15,13)(12,17,18,16)

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7610897);