Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #42996 |
| automorphism | ? |
| homology | Abelianization (H1) = C26 · Schur multiplier (H2) = C212 |
| direct product | C2×#43635 |
| 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 | |
4986 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)(5,9)(6,10)(7,8)(12,15)(16,18)(19,20,23,24)(21,25,27,28)(22,30,31,26)(29,34,33,32);(4,8)(5,10)(20,24)(21,27)(22,31)(32,34);(11,12)(13,16)(14,15)(17,18)(19,20)(21,28)(22,26)(23,24)(25,27)(29,34)(30,31)(32,33);(3,5,7,10)(4,9,8,6)(11,13,14,17)(12,16,15,18)(19,21)(20,25)(22,29)(23,27)(24,28)(26,32)(30,34)(31,33);(3,6,7,9)(4,10,8,5)(19,22,23,31)(20,26,24,30)(21,29,27,33)(25,32,28,34)116 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7574976);