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×#62962 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C26 |
| commutator, Frattini, cosocle | C22 | C27 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C22 ▷ C1 | |
32100 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);(3,4,8,11)(5,12,6,9)(7,15,16,10)(13,17,14,18)(21,27)(25,28)(29,33)(32,34);(3,4)(5,12)(6,9)(7,10)(8,11)(13,18)(14,17)(15,16)(19,20,24,22)(21,25,27,28)(23,26,31,30)(29,32,33,34);(3,5,8,6)(4,9,11,12)(7,13,16,14)(10,17,15,18)(19,21)(20,25)(22,28)(23,29)(24,27)(26,32)(30,34)(31,33);(3,4)(5,9)(6,12)(7,10)(8,11)(13,17)(14,18)(15,16)(19,22,24,20)(21,28,27,25)(23,26,31,30)(29,32,33,34);(3,6,8,5)(4,9,11,12)(7,14,16,13)(10,17,15,18)(19,23,24,31)(20,26,22,30)(21,29,27,33)(25,32,28,34);(3,7)(4,10)(5,13)(6,14)(8,16)(9,17)(11,15)(12,18)(19,21)(20,25)(22,28)(23,29)(24,27)(26,32)(30,34)(31,33)146 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 10493503);