Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #47826 |
| automorphism | ? |
| homology | Abelianization (H1) = C26 · Schur multiplier (H2) = C212 |
| direct product | C2×#51483 |
| 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 | |
4854 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);(3,4)(5,8)(6,7)(9,10)(11,12,15,13)(14,17,18,16)(19,20,24,27)(21,28,29,25)(22,23,26,33)(30,31,32,34);(4,6)(8,9)(11,13,15,12)(14,17,18,16)(21,29)(22,26)(23,33)(25,28);(3,4,7,6)(5,9,10,8)(11,14)(12,16)(13,17)(15,18)(19,21,24,29)(20,25,27,28)(22,32,26,30)(23,31,33,34);(3,5,7,10)(4,8,6,9)(19,22,24,26)(20,23,27,33)(21,30,29,32)(25,34,28,31);(3,6,7,4)(5,9,10,8)(19,23)(20,26)(21,31)(22,27)(24,33)(25,30)(28,32)(29,34)104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7614539);