Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #52308 |
| automorphism | ? |
| homology | Abelianization (H1) = C26 · Schur multiplier (H2) = C212 |
| direct product | C2×#52368 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C24 | C25 |
| commutator, Frattini, cosocle | C23 | C26 |
| prime radical | C26 | C23 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
5306 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)(11,12,14,16)(13,17,18,15)(19,20)(21,26)(22,30)(23,31)(24,25)(27,28)(29,33)(32,34);(3,4)(5,8)(6,7)(9,10)(11,12,14,16)(13,15,18,17)(20,25)(22,31)(26,33)(27,32);(4,7)(8,10)(11,12)(13,15)(14,16)(17,18)(19,21,24,29)(20,26,25,33)(22,32,31,27)(23,28,30,34);(3,4)(5,8)(6,7)(9,10)(11,13)(12,15)(14,18)(16,17)(19,22)(20,23)(21,27)(24,31)(25,30)(26,34)(28,33)(29,32);(3,5)(4,8)(6,9)(7,10)(19,23)(20,22)(21,28)(24,30)(25,31)(26,32)(27,33)(29,34)92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7999318);