Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #36702 |
| automorphism | ? |
| homology | Abelianization (H1) = C26 · Schur multiplier (H2) = C212 |
| direct product | C2×#36733 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C24 | C25 |
| commutator, Frattini, cosocle | C23 | C26 |
| prime radical | C22×#120 | C22 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
6200 conjugacy classes of subgroups.
| minimal degree | μ(G) = 34 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 34 points:
(1,2);(5,9)(8,10)(13,17)(14,18)(19,20)(21,27)(22,26)(23,24)(25,29)(28,33)(30,31)(32,34);(4,7)(8,10)(11,12,15,16)(13,18,17,14)(20,24)(25,27)(26,31)(32,33);(3,4)(5,8)(6,7)(9,10)(11,13)(12,14)(15,17)(16,18)(19,20)(21,25)(22,26)(23,24)(27,29)(28,32)(30,31)(33,34);(3,4)(5,8)(6,7)(9,10)(11,12)(13,14)(15,16)(17,18)(19,21)(20,25)(22,28)(23,29)(24,27)(26,32)(30,34)(31,33);(3,5)(4,8)(6,9)(7,10)(11,14)(12,13)(15,18)(16,17)(19,22,23,30)(20,26,24,31)(21,28,29,34)(25,32,27,33)104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7541350);