Finite Group Databasebeta

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Cayley diagram

C2×#52497 #8092189

order512
nilpotency class2
isoclinismisoclinic class of #52308
automorphism?
homologyAbelianization (H1) = C26 · Schur multiplier (H2) = C212

Aliases

direct productC2×#52497

Subgroups

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

5634 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);(3,4,7,8)(5,10,9,6)(13,17)(16,18)(19,20,23,25)(21,26,24,22);(11,12)(13,16)(14,15)(17,18)(19,21)(20,22)(23,24)(25,26);(4,8)(5,9)(12,15)(16,18)(19,20,23,25)(21,22,24,26);(3,5,7,9)(4,6,8,10)(11,12)(13,16)(14,15)(17,18)(19,20)(21,26)(22,24)(23,25);(3,6)(4,9)(5,8)(7,10)(11,13)(12,16)(14,17)(15,18)(19,22,23,26)(20,24,25,21)

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, 7999385);