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×#44819 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C24 | C25 |
| commutator, Frattini, cosocle | C23 | C26 |
| prime radical | C2×#2703 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
5528 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);(3,4,7,9)(5,8,10,6)(11,12,14,16)(13,15,17,18)(22,30)(26,31)(28,34)(32,33);(3,5,7,10)(4,6,9,8)(20,24)(22,30)(25,27)(28,34);(11,12)(13,15)(14,16)(17,18)(19,20,23,24)(21,27,29,25)(22,26,30,31)(28,33,34,32);(3,5)(4,8)(6,9)(7,10)(19,21,23,29)(20,25,24,27)(22,28,30,34)(26,32,31,33);(3,6,7,8)(4,5,9,10)(11,13,14,17)(12,15,16,18)(19,22)(20,26)(21,28)(23,30)(24,31)(25,32)(27,33)(29,34)116 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7576398);