Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #40484 |
| automorphism | ? |
| homology | Abelianization (H1) = C26 · Schur multiplier (H2) = C212 |
| direct product | C2×#41896 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C24 | C25 |
| commutator, Frattini, cosocle | C23 | C26 |
| prime radical | C22×#120 | C22 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
5324 conjugacy classes of subgroups.
| minimal degree | μ(G) = 34 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 34 points:
(1,2);(3,4,7,6)(5,9,10,8)(11,12,14,16)(13,17,18,15)(20,24)(25,27)(26,31)(32,33);(11,13,14,18)(12,15,16,17)(20,24)(22,30)(25,27)(28,34);(3,5,7,10)(4,8,6,9)(11,13)(12,15)(14,18)(16,17)(19,20,23,24)(21,27,29,25)(22,26,30,31)(28,33,34,32);(3,4,7,6)(5,8,10,9)(11,12)(13,15)(14,16)(17,18)(19,21,23,29)(20,25,24,27)(22,28,30,34)(26,32,31,33);(3,6,7,4)(5,9,10,8)(19,22,23,30)(20,26,24,31)(21,28,29,34)(25,32,27,33)104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7547417);