Finite Group Databasebeta

← Prev (#7703402) Next (#7703404) →

Cayley diagram

C2×#48556 #7703403

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

Aliases

direct productC2×#48556

Subgroups

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

6652 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 26
transitive degreeμt(G) = 128

A minimal faithful permutation representation — 6 generators on 26 points:

(1,2);(3,4,7,5)(6,9,10,8)(11,12)(13,15)(14,18)(16,17);(4,5)(8,9)(11,13)(12,15)(14,16)(17,18)(19,20)(21,25)(22,26)(23,24);(12,17)(13,14)(20,24)(21,26);(3,5,7,4)(6,9,10,8)(11,14,16,13)(12,18,17,15)(19,21)(20,22)(23,26)(24,25);(3,6)(4,8)(5,9)(7,10)(11,15,16,18)(12,13,17,14)(19,22)(20,21)(23,25)(24,26)

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, 7610599);