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×#40541 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C24 | C25 |
| commutator, Frattini, cosocle | C23 | C26 |
| prime radical | C26 | C23 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
5942 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);(11,12)(13,14)(15,16)(17,18)(19,20,24,25)(21,26,30,32)(22,29,33,27)(23,28,34,31);(3,4)(5,9)(6,7)(8,10)(11,13,15,17)(12,14,16,18)(20,25)(23,34)(26,32)(27,29);(4,7)(8,9)(12,16)(13,17)(19,21)(20,26)(22,31)(23,29)(24,30)(25,32)(27,34)(28,33);(3,4)(5,8)(6,7)(9,10)(11,12)(13,14)(15,16)(17,18)(19,22,24,33)(20,27,25,29)(21,28,30,31)(23,32,34,26);(3,5)(4,8)(6,10)(7,9)(11,14)(12,13)(15,18)(16,17)(19,23)(20,28)(21,29)(22,32)(24,34)(25,31)(26,33)(27,30)104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7546100);