Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #45438 |
| automorphism | ? |
| homology | Abelianization (H1) = C26 · Schur multiplier (H2) = C212 |
| direct product | C2×#46116 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C24 | C25 |
| commutator, Frattini, cosocle | C23 | C26 |
| prime radical | C25 | C24 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4790 conjugacy classes of subgroups.
| minimal degree | μ(G) = 34 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 34 points:
(1,2);(5,9)(8,10)(11,12,15,18)(13,16,14,17)(19,20)(21,22)(23,31)(24,25)(26,30)(27,32)(28,34)(29,33);(4,7)(5,9)(20,25)(21,26)(23,32)(29,34);(3,4)(5,8)(6,7)(9,10)(11,12)(13,16)(14,17)(15,18)(19,21,24,26)(20,22,25,30)(23,28,32,33)(27,29,31,34);(3,5,6,9)(4,8,7,10)(11,13,15,14)(12,16,18,17)(19,22)(20,26)(21,25)(23,29)(24,30)(27,33)(28,31)(32,34);(3,4)(5,8)(6,7)(9,10)(11,14,15,13)(12,17,18,16)(19,23)(20,27)(21,28)(22,29)(24,32)(25,31)(26,33)(30,34)104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7609779);