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×#62971 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C26 |
| commutator, Frattini, cosocle | C22 | C27 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C22 ▷ C1 | |
31672 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)(9,12)(13,17)(16,18)(19,20,24,21)(22,27,30,25)(23,26,31,28)(29,33,34,32);(3,4,8,6)(5,9,11,12)(7,10,15,14)(13,16,17,18)(20,21)(25,27)(26,28)(32,33);(3,5)(4,9)(6,12)(7,13)(8,11)(10,16)(14,18)(15,17)(19,21,24,20)(22,27,30,25)(23,28,31,26)(29,33,34,32);(3,6,8,4)(5,12,11,9)(7,10,15,14)(13,16,17,18)(19,22)(20,25)(21,27)(23,29)(24,30)(26,32)(28,33)(31,34);(3,7,8,15)(4,10,6,14)(5,13,11,17)(9,16,12,18);(3,5)(4,9)(6,12)(7,13)(8,11)(10,16)(14,18)(15,17)(19,23,24,31)(20,26,21,28)(22,29,30,34)(25,32,27,33)146 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 10493512);