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×#37886 |
| 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 | |
5006 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);(3,4,7,9)(5,6,8,10)(13,17)(15,18)(19,20,24,25)(21,28,26,22)(23,27,31,32)(29,34,33,30);(3,5)(4,6)(7,8)(9,10)(11,12,14,16)(13,15,17,18)(20,25)(21,26)(27,32)(29,33);(3,6)(4,8)(5,9)(7,10)(11,13,14,17)(12,15,16,18)(19,21,24,26)(20,22,25,28)(23,29,31,33)(27,30,32,34);(19,22)(20,26)(21,25)(23,30)(24,28)(27,33)(29,32)(31,34);(3,6)(4,8)(5,9)(7,10)(19,23,24,31)(20,27,25,32)(21,29,26,33)(22,30,28,34)116 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7542500);