Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #37644 |
| automorphism | ? |
| homology | Abelianization (H1) = C26 · Schur multiplier (H2) = C212 |
| direct product | C2×#38063 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C24 | C25 |
| commutator, Frattini, cosocle | C23 | C26 |
| prime radical | C23×#41 | C22 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
5766 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);(13,17)(15,18)(19,20,24,23)(21,27,29,25)(22,30,31,26)(28,32,34,33);(4,7)(5,9)(11,12,14,16)(13,18,17,15)(20,23)(25,27)(26,30)(32,33);(3,4,6,7)(5,8,9,10)(11,13,14,17)(12,15,16,18)(19,21)(20,25)(22,28)(23,27)(24,29)(26,32)(30,33)(31,34);(19,22,24,31)(20,26,23,30)(21,28,29,34)(25,32,27,33);(3,5)(4,8)(6,9)(7,10)(11,12)(13,15)(14,16)(17,18)(19,23,24,20)(21,27,29,25)(22,30,31,26)(28,33,34,32)116 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7542900);