Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #24987 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
| direct product | C2×#24987 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2420 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | C22×AGL(1,ℤ/8) | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1824 conjugacy classes of subgroups.
| minimal degree | μ(G) = 34 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 5 generators on 34 points:
(1,2);(4,9)(5,13)(6,17)(7,19)(11,26)(12,22)(15,16)(18,30)(20,24)(23,31)(27,28)(33,34);(3,4,7,9,8,12,19,22)(5,10,13,24,16,25,15,20)(6,11,17,21,18,26,30,32)(14,23,27,33,29,34,28,31);(3,5)(4,10)(6,14)(7,15)(8,16)(9,20)(11,23)(12,25)(13,19)(17,28)(18,29)(21,31)(22,24)(26,34)(27,30)(32,33);(3,6,7,17,8,18,19,30)(4,11,9,21,12,26,22,32)(5,14,15,28,16,29,13,27)(10,23,20,31,25,34,24,33)80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6354244);