Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #62892 |
| automorphism | ? |
| homology | Abelianization (H1) = C27 · Schur multiplier (H2) = C219 |
| direct product | C2×#62951 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C26 |
| commutator, Frattini, cosocle | C22 | C27 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C22 ▷ C1 | |
32138 conjugacy classes of subgroups.
| minimal degree | μ(G) = 34 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 7 generators on 34 points:
(1,2);(4,8)(9,15)(10,16)(17,18)(19,20,26,24)(21,30,29,25)(22,32,34,27)(23,31,33,28);(19,21)(20,25)(22,28)(23,27)(24,30)(26,29)(31,34)(32,33);(3,4)(5,9)(6,10)(7,8)(11,17)(12,15)(13,18)(14,16)(19,22)(20,27)(21,31)(23,30)(24,32)(25,33)(26,34)(28,29);(3,5,7,12)(4,9,8,15)(6,13,14,11)(10,18,16,17)(19,23,26,33)(20,28,24,31)(21,27,29,32)(22,30,34,25);(3,6,7,14)(4,10,8,16)(5,11,12,13)(9,17,15,18)(19,24,26,20)(21,30,29,25)(22,32,34,27)(23,28,33,31);(3,4)(5,9)(6,10)(7,8)(11,17)(12,15)(13,18)(14,16)(19,25,26,30)(20,29,24,21)(22,33,34,23)(27,28,32,31)146 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 10493491);