Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #53390 |
| automorphism | ? |
| homology | Abelianization (H1) = C26 · Schur multiplier (H2) = C212 |
| direct product | C2×#53581 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C24 | C25 |
| commutator, Frattini, cosocle | C23 | C26 |
| prime radical | C2×#2670 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
5762 conjugacy classes of subgroups.
| minimal degree | μ(G) = 34 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 34 points:
(1,2);(3,4)(5,9)(6,7)(8,10)(19,20,24,27)(21,25,22,29)(23,26,31,34)(28,32,30,33);(4,7)(8,9)(11,12)(13,17)(14,15)(16,18)(19,21,24,22)(20,25,27,29)(23,30,31,28)(26,33,34,32);(12,15)(16,17)(19,22,24,21)(20,25,27,29)(23,30,31,28)(26,32,34,33);(3,4)(5,8)(6,7)(9,10)(11,12)(13,16)(14,15)(17,18)(19,20)(21,25)(22,29)(23,26)(24,27)(28,32)(30,33)(31,34);(3,5)(4,8)(6,10)(7,9)(11,13)(12,16)(14,18)(15,17)(19,23)(20,26)(21,28)(22,30)(24,31)(25,32)(27,34)(29,33)92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8006677);