Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #24985 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
| direct product | C2×#24985 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #555 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| socle | C22 | #2420 |
| prime radical | C2×#3067 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1838 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);(3,4)(5,17)(6,21)(7,11)(8,15)(9,14)(10,16)(12,24)(13,20)(18,33)(19,28)(22,30)(23,29)(25,31)(26,32)(27,34);(3,5,9,19,10,20,8,7)(4,11,15,13,16,28,14,17)(6,22,24,18,26,23,25,34)(12,29,21,33,31,30,32,27);(3,6)(4,12)(5,18)(7,23)(8,24)(9,25)(10,26)(11,27)(13,30)(14,21)(15,32)(16,31)(17,29)(19,22)(20,34)(28,33);(3,7,8,20,10,19,9,5)(4,13,14,11,16,17,15,28)(6,23,24,34,26,22,25,18)(12,30,21,27,31,29,32,33)92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6354250);