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×#46190 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C24 | C25 |
| commutator, Frattini, cosocle | C23 | C26 |
| prime radical | C23×D4 | C23 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4990 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);(3,4,8,9)(5,6,7,10)(12,15)(16,18)(19,20,24,27)(21,22,28,25)(23,26,31,33)(29,30,34,32);(4,9)(5,7)(21,28)(22,25)(23,31)(26,33);(3,5,8,7)(4,10,9,6)(11,12)(13,16)(14,15)(17,18)(19,21)(20,25)(22,27)(23,29)(24,28)(26,32)(30,33)(31,34);(3,6)(4,5)(7,9)(8,10)(19,22,24,25)(20,21,27,28)(23,30,31,32)(26,29,33,34);(3,7,8,5)(4,6,9,10)(11,13,14,17)(12,16,15,18)(19,23,24,31)(20,26,27,33)(21,29,28,34)(22,30,25,32)104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7609955);