Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #52308 |
| automorphism | ? |
| homology | Abelianization (H1) = C26 · Schur multiplier (H2) = C212 |
| direct product | C2×#52875 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C24 | C25 |
| commutator, Frattini, cosocle | C23 | C26 |
| prime radical | C2×#2813 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
5066 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);(3,4,6,7)(5,9,10,8)(12,17)(13,15)(19,20)(21,29)(22,27)(23,28)(24,25)(26,32)(30,31)(33,34);(11,12,16,17)(13,14,15,18)(20,25)(22,28)(26,29)(30,34);(4,7)(8,9)(11,13,16,15)(12,18,17,14)(19,21,24,32)(20,26,25,29)(22,30,28,34)(23,31,27,33);(3,4)(5,8)(6,7)(9,10)(11,14,16,18)(12,13,17,15)(19,22,24,28)(20,27,25,23)(21,30,32,34)(26,33,29,31);(3,5,6,10)(4,8,7,9)(11,15,16,13)(12,18,17,14)(19,23,24,27)(20,28,25,22)(21,31,32,33)(26,34,29,30)92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7999744);