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×#40439 |
| 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 | |
5100 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);(12,15)(16,18)(19,20,25,28)(21,29,24,26)(22,32,27,23)(30,34,33,31);(4,8)(6,9)(12,15)(13,17)(19,21,25,24)(20,26,28,29)(22,33,27,30)(23,31,32,34);(3,4)(5,9)(6,10)(7,8)(11,12,14,15)(13,16,17,18)(19,22,25,27)(20,23,28,32)(21,30,24,33)(26,34,29,31);(3,5,7,10)(4,9,8,6)(19,23)(20,27)(21,31)(22,28)(24,34)(25,32)(26,30)(29,33);(3,6,7,9)(4,10,8,5)(11,13)(12,16)(14,17)(15,18)(19,24,25,21)(20,26,28,29)(22,33,27,30)(23,34,32,31)116 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7545060);