Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #62817 |
| automorphism | ? |
| homology | Abelianization (H1) = C27 · Schur multiplier (H2) = C219 |
| direct product | C2×#62838 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C26 |
| commutator, Frattini, cosocle | C22 | C27 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C22 ▷ C1 | |
31954 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)(5,11)(6,14)(15,18)(21,27)(24,31)(28,33)(32,34);(3,4,9,8)(5,12,11,7)(6,10,14,16)(13,18,17,15)(19,20,23,26)(21,24,27,31)(22,29,30,25)(28,34,33,32);(3,5)(4,7)(6,13)(8,12)(9,11)(10,15)(14,17)(16,18)(19,21)(20,24)(22,28)(23,27)(25,32)(26,31)(29,34)(30,33);(3,6)(4,10)(5,13)(7,15)(8,16)(9,14)(11,17)(12,18);(3,7,9,12)(4,11,8,5)(6,15,14,18)(10,17,16,13)(19,22,23,30)(20,25,26,29)(21,28,27,33)(24,32,31,34);(3,8,9,4)(5,12,11,7)(6,16,14,10)(13,18,17,15)(19,21)(20,24)(22,28)(23,27)(25,32)(26,31)(29,34)(30,33)152 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 10493384);