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×#41357 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C24 | C25 |
| commutator, Frattini, cosocle | C23 | C26 |
| prime radical | C25 | C24 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4800 conjugacy classes of subgroups.
| minimal degree | μ(G) = 34 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 34 points:
(1,2);(11,12,15,14)(13,17,18,16)(19,20,24,25)(21,29,31,26)(22,32,33,27)(23,28,34,30);(3,4)(5,9)(6,7)(8,10)(11,12,15,14)(13,16,18,17)(20,25)(23,34)(26,29)(27,32);(4,7)(8,9)(11,13,15,18)(12,16,14,17)(19,21,24,31)(20,26,25,29)(22,28,33,30)(23,27,34,32);(3,4)(5,8)(6,7)(9,10)(11,12,15,14)(13,16,18,17)(19,22,24,33)(20,27,25,32)(21,30,31,28)(23,29,34,26);(3,5)(4,8)(6,10)(7,9)(11,14,15,12)(13,17,18,16)(19,23)(20,28)(21,27)(22,29)(24,34)(25,30)(26,33)(31,32)104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7546445);