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×#62840 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C26 |
| commutator, Frattini, cosocle | C22 | C27 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C22 ▷ C1 | |
32252 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)(9,14)(10,15)(16,18)(19,20)(21,29)(22,28)(23,27)(24,25)(26,30)(31,33)(32,34);(6,13)(10,15)(11,17)(16,18)(20,25)(21,30)(27,28)(31,32);(3,4)(5,9)(6,10)(7,8)(11,16)(12,14)(13,15)(17,18)(19,21)(20,26)(22,31)(23,32)(24,30)(25,29)(27,33)(28,34);(3,5,7,12)(4,9,8,14)(6,11,13,17)(10,16,15,18)(19,22,24,23)(20,27,25,28)(21,31,30,32)(26,33,29,34);(3,6,7,13)(4,10,8,15)(5,11,12,17)(9,16,14,18)(19,20)(21,26)(22,27)(23,28)(24,25)(29,30)(31,33)(32,34);(3,4)(5,9)(6,10)(7,8)(11,16)(12,14)(13,15)(17,18)(19,23,24,22)(20,28,25,27)(21,32,30,31)(26,34,29,33)152 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 10493387);