Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #53390 |
| automorphism | ? |
| homology | Abelianization (H1) = C26 · Schur multiplier (H2) = C212 |
| direct product | C2×#57207 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C24 | C25 |
| commutator, Frattini, cosocle | C23 | C26 |
| prime radical | C2×#2430 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
5798 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)(5,8)(6,7)(9,10)(19,20)(21,27)(22,30)(23,29)(24,25)(26,31)(28,33)(32,34);(3,5,7,10)(4,8,6,9)(11,12,15,17)(13,14,18,16)(20,25)(22,31)(23,27)(28,32);(4,6)(8,9)(11,13,15,18)(12,16,17,14)(19,21)(20,23)(22,32)(24,29)(25,27)(26,34)(28,31)(30,33);(3,6,7,4)(5,9,10,8)(11,13)(12,14)(15,18)(16,17)(19,22)(20,26)(21,28)(23,33)(24,31)(25,30)(27,34)(29,32);(3,5)(4,8)(6,9)(7,10)(11,14)(12,13)(15,16)(17,18)(19,23)(20,21)(22,33)(24,27)(25,29)(26,28)(30,32)(31,34)92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8007731);