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×#62950 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C26 |
| commutator, Frattini, cosocle | C22 | C27 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C22 ▷ C1 | |
31358 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)(21,27)(22,26)(23,24)(25,29)(28,33)(30,31)(32,34);(20,24)(25,27)(26,31)(32,33);(3,4)(5,9)(6,10)(7,8)(11,17)(12,15)(13,18)(14,16)(19,20)(21,25)(22,26)(23,24)(27,29)(28,32)(30,31)(33,34);(3,5,7,12)(4,9,8,15)(6,13,14,11)(10,18,16,17)(19,21)(20,25)(22,28)(23,29)(24,27)(26,32)(30,34)(31,33);(3,6,7,14)(4,10,8,16)(5,11,12,13)(9,17,15,18);(3,4)(5,9)(6,10)(7,8)(11,17)(12,15)(13,18)(14,16)(19,22,23,30)(20,26,24,31)(21,28,29,34)(25,32,27,33)146 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 10493490);