Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #47826 |
| automorphism | ? |
| homology | Abelianization (H1) = C26 · Schur multiplier (H2) = C212 |
| direct product | C2×#49755 |
| 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 | |
5240 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);(3,4,8,10)(5,9,6,7)(19,20,24,21)(22,25,30,27)(23,28,31,26)(29,33,34,32);(3,5)(4,9)(6,8)(7,10)(12,15)(13,17)(20,21)(23,31)(25,27)(29,34);(4,10)(5,6)(11,12,14,15)(13,16,17,18)(19,21,24,20)(22,25,30,27)(23,28,31,26)(29,32,34,33);(3,6,8,5)(4,7,10,9)(11,12)(13,16)(14,15)(17,18)(19,22)(20,25)(21,27)(23,29)(24,30)(26,32)(28,33)(31,34);(3,7)(4,5)(6,10)(8,9)(11,13)(12,16)(14,17)(15,18)(19,23,24,31)(20,26,21,28)(22,29,30,34)(25,32,27,33)104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7612529);