Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #35680 |
| automorphism | ? |
| homology | Abelianization (H1) = C26 · Schur multiplier (H2) = C212 |
| direct product | C2×#35789 |
| 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 | |
5214 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);(3,4,6,8)(5,7,9,10)(11,12,15,14)(13,17,18,16);(11,13,15,18)(12,16,14,17)(20,24)(25,31)(26,32)(33,34);(3,4)(5,7)(6,8)(9,10)(19,20)(21,25)(22,26)(23,24)(27,33)(28,31)(29,34)(30,32);(3,5,6,9)(4,7,8,10)(11,14,15,12)(13,17,18,16)(19,21,23,28)(20,25,24,31)(22,29,30,27)(26,34,32,33);(19,22,23,30)(20,26,24,32)(21,27,28,29)(25,33,31,34)128 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7536669);