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×#62959 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C26 |
| commutator, Frattini, cosocle | C22 | C27 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C22 ▷ C1 | |
31352 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)(20,24)(22,30)(25,27)(28,34);(3,4,8,6)(5,9,11,12)(7,10,15,14)(13,16,17,18)(19,20)(21,27)(22,26)(23,24)(25,29)(28,33)(30,31)(32,34);(3,5)(4,9)(6,12)(7,13)(8,11)(10,16)(14,18)(15,17)(19,21,23,29)(20,25,24,27)(22,28,30,34)(26,32,31,33);(3,6,8,4)(5,12,11,9)(7,10,15,14)(13,16,17,18)(19,22,23,30)(20,26,24,31)(21,28,29,34)(25,32,27,33);(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)146 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 10493500);