Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #18746 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C29 |
| direct product | C2×#18789 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C2×#120 |
| commutator, Frattini, cosocle | C42 | C25 |
| prime radical | C22×#41 | D4 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C42 ▷ C22 ▷ C1 | |
2982 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);(3,4,7,10)(5,11,13,18)(6,14,16,9)(8,15,17,12)(19,20,25,30)(21,27,32,33)(22,26,31,34)(23,29,24,28);(3,5,7,13)(4,8,10,17)(6,15,16,12)(9,18,14,11)(19,21,25,32)(20,26,30,34)(22,23,31,24)(27,28,33,29);(19,22)(20,27)(21,23)(24,32)(25,31)(26,28)(29,34)(30,33);(19,23,25,24)(20,28,30,29)(21,31,32,22)(26,33,34,27)86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6255902);