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×#43516 |
| 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 | |
5072 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);(3,4,8,7)(5,10,6,9)(11,12,15,13)(14,17,18,16)(19,20,24,22)(21,27,28,25)(23,30,31,26)(29,32,34,33);(3,5,8,6)(4,9,7,10)(11,13,15,12)(14,17,18,16)(20,22)(21,28)(23,31)(32,33);(3,6,8,5)(4,9,7,10)(11,14)(12,16)(13,17)(15,18)(19,21,24,28)(20,25,22,27)(23,29,31,34)(26,32,30,33);(3,7,8,4)(5,9,6,10)(19,22,24,20)(21,25,28,27)(23,30,31,26)(29,32,34,33);(19,23)(20,26)(21,29)(22,30)(24,31)(25,32)(27,33)(28,34)116 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7574949);