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×#44669 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C24 | C25 |
| commutator, Frattini, cosocle | C23 | C26 |
| prime radical | C22×#121 | C22 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
5382 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)(5,8)(6,9)(7,10)(13,17)(16,18)(19,20,23,24)(21,27,28,25)(22,26,30,32)(29,34,31,33);(3,4,7,10)(5,8,6,9)(11,12,15,14)(13,16,17,18)(20,24)(21,28)(22,30)(33,34);(3,5,7,6)(4,8,10,9)(11,13,15,17)(12,16,14,18)(19,21,23,28)(20,25,24,27)(22,31,30,29)(26,34,32,33);(3,4)(5,8)(6,9)(7,10)(19,22)(20,26)(21,29)(23,30)(24,32)(25,33)(27,34)(28,31);(3,6,7,5)(4,9,10,8)(11,14,15,12)(13,18,17,16)(19,20)(21,25)(22,26)(23,24)(27,28)(29,33)(30,32)(31,34)116 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7576188);