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×#40330 |
| 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 | |
4850 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)(21,28)(22,25)(23,31)(24,27)(26,32)(29,33)(30,34);(4,8)(6,9)(19,21,24,22)(20,25,27,28)(23,29,32,30)(26,33,31,34);(3,4)(5,9)(6,10)(7,8)(11,13,15,18)(12,16,14,17)(19,22,24,21)(20,25,27,28)(23,30,32,29)(26,33,31,34);(3,5,7,10)(4,9,8,6)(19,23)(20,26)(21,29)(22,30)(24,32)(25,33)(27,31)(28,34);(3,6,7,9)(4,10,8,5)(11,14,15,12)(13,17,18,16)(19,20)(21,28)(22,25)(23,26)(24,27)(29,34)(30,33)(31,32)116 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7544738);