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×#62939 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C26 |
| commutator, Frattini, cosocle | C22 | C27 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C22 ▷ C1 | |
31650 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);(5,11)(9,13)(12,17)(16,18)(19,20,24,27)(21,22,29,25)(23,26,31,33)(28,30,34,32);(3,4,8,7)(5,9,11,13)(6,10,14,15)(12,16,17,18)(19,21,24,29)(20,25,27,22)(23,28,31,34)(26,32,33,30);(3,5)(4,9)(6,12)(7,13)(8,11)(10,16)(14,17)(15,18)(19,21)(20,22)(23,28)(24,29)(25,27)(26,30)(31,34)(32,33);(3,6)(4,10)(5,12)(7,15)(8,14)(9,16)(11,17)(13,18)(19,22)(20,21)(23,30)(24,25)(26,28)(27,29)(31,32)(33,34);(3,7,8,4)(5,13,11,9)(6,10,14,15)(12,16,17,18);(3,5)(4,9)(6,12)(7,13)(8,11)(10,16)(14,17)(15,18)(19,23,24,31)(20,26,27,33)(21,28,29,34)(22,30,25,32)146 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 10493478);