Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #35680 |
| automorphism | ? |
| homology | Abelianization (H1) = C26 · Schur multiplier (H2) = C212 |
| direct product | C2×#36072 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C24 | C25 |
| commutator, Frattini, cosocle | C23 | C26 |
| prime radical | C23×D4 | C23 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
5318 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);(3,4,6,8)(5,7,9,10)(11,12,15,14)(13,17,18,16)(19,20)(21,23)(22,27)(24,25)(26,28)(29,32)(30,34)(31,33);(3,4)(5,7)(6,8)(9,10)(11,13,15,18)(12,16,14,17)(20,25)(21,28)(27,33)(29,34);(3,4)(5,7)(6,8)(9,10)(11,12,15,14)(13,16,18,17)(19,21)(20,26)(22,29)(23,25)(24,28)(27,30)(31,34)(32,33);(11,12,15,14)(13,16,18,17)(19,22)(20,27)(21,29)(23,32)(24,31)(25,33)(26,30)(28,34);(3,5,6,9)(4,7,8,10)(11,14,15,12)(13,17,18,16)(19,23,24,26)(20,21,25,28)(22,30,31,32)(27,34,33,29)128 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7536735);