Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #52308 |
| automorphism | ? |
| homology | Abelianization (H1) = C26 · Schur multiplier (H2) = C212 |
| direct product | C2×#52485 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C24 | C25 |
| commutator, Frattini, cosocle | C23 | C26 |
| prime radical | C2×#2714 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
5520 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)(5,7)(6,9)(8,10)(13,17)(16,18)(19,20,24,27)(21,25,29,23)(22,30,33,26)(28,32,31,34);(11,12)(13,16)(14,15)(17,18)(19,21,24,29)(20,23,27,25)(22,31,33,28)(26,34,30,32);(4,5)(8,9)(12,15)(16,18)(19,21)(20,25)(22,28)(23,27)(24,29)(26,34)(30,32)(31,33);(3,5,7,4)(6,9,10,8)(11,12)(13,16)(14,15)(17,18)(19,22,24,33)(20,26,27,30)(21,28,29,31)(23,32,25,34);(3,6)(4,8)(5,9)(7,10)(11,13)(12,16)(14,17)(15,18)(19,23,24,25)(20,21,27,29)(22,32,33,34)(26,28,30,31)92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7999357);