Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #9263 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C26 |
| direct product | C2×#9270 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | #386 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C2×C42 | C24 |
| prime radical | C24 | C4×D4 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
1100 conjugacy classes of subgroups.
| minimal degree | μ(G) = 34 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 4 generators on 34 points:
(1,2);(4,8)(5,12)(6,9)(11,16)(13,14)(15,17)(19,20,23,25,24,28,22,27)(21,29,31,26,32,33,30,34);(3,4,9,16,7,11,6,8)(5,10,12,17,14,18,13,15)(20,25)(22,23)(26,29)(27,28)(30,31)(33,34);(3,5)(4,10)(6,13)(7,14)(8,15)(9,12)(11,18)(16,17)(19,21,24,32)(20,26,28,34)(22,31,23,30)(25,29,27,33)86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 105791);