Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #35680 |
| automorphism | ? |
| homology | Abelianization (H1) = C26 · Schur multiplier (H2) = C212 |
| direct product | C2×#35805 |
| 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 | |
5318 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)(5,10,9,6)(12,16)(17,18);(3,5)(4,6)(7,10)(8,9)(11,12)(13,17)(14,18)(15,16)(20,24)(25,31)(26,32)(33,34);(3,6,8,10)(4,9,7,5)(11,13,15,14)(12,17,16,18)(19,20)(21,25)(22,26)(23,24)(27,33)(28,31)(29,34)(30,32);(3,4,8,7)(5,6,9,10)(19,21,23,28)(20,25,24,31)(22,29,30,27)(26,34,32,33);(3,7,8,4)(5,10,9,6)(11,14,15,13)(12,18,16,17)(19,22,23,30)(20,26,24,32)(21,27,28,29)(25,33,31,34)128 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7536712);