Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #19651 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27×C4 |
| direct product | C2×#19659 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C2×#127 |
| commutator, Frattini, cosocle | C42 | C25 |
| prime radical | C2×#532 | C22 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C42 ▷ C22 ▷ C1 | |
2028 conjugacy classes of subgroups.
| minimal degree | μ(G) = 34 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 34 points:
(1,2);(3,4)(5,11)(6,8)(7,15)(9,13)(10,14)(12,17)(16,18)(19,20,24,27)(21,28,26,23)(22,30,31,29)(25,33,34,32);(4,8)(9,16)(10,15)(11,17)(19,21,23,20,24,26,28,27)(22,29,32,25,31,30,33,34);(3,5,7,13,6,12,14,18)(4,9,10,11,8,16,15,17)(19,22,23,32,24,31,28,33)(20,25,26,30,27,34,21,29);(22,31)(25,34)(29,30)(32,33)104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6264348);