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×#62911 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C26 |
| commutator, Frattini, cosocle | C22 | C27 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C22 ▷ C1 | |
31704 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);(5,12)(9,16)(13,17)(14,18)(20,22)(25,27)(26,30)(32,33);(3,4,8,11)(5,9,12,16)(6,7,10,15)(13,14,17,18)(19,20,24,22)(21,27,29,25)(23,26,31,30)(28,33,34,32);(3,5)(4,9)(6,13)(7,14)(8,12)(10,17)(11,16)(15,18)(19,21,24,29)(20,25,22,27)(23,28,31,34)(26,32,30,33);(3,6)(4,7)(5,13)(8,10)(9,14)(11,15)(12,17)(16,18)(19,22,24,20)(21,27,29,25)(23,30,31,26)(28,33,34,32);(3,7)(4,10)(5,14)(6,11)(8,15)(9,17)(12,18)(13,16)(19,23,24,31)(20,26,22,30)(21,28,29,34)(25,32,27,33);(3,5)(4,9)(6,13)(7,14)(8,12)(10,17)(11,16)(15,18)146 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 10493448);