Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #24398 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C29 |
| direct product | C2×#24775 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C2×#121 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | C22×AGL(1,ℤ/8) | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2548 conjugacy classes of subgroups.
| minimal degree | μ(G) = 38 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 38 points:
(1,2);(4,5)(7,8,12,17,11,16,13,18)(9,19,22,34,21,14,28,30)(10,23,26,32,25,33,27,15)(20,36,24,29,37,31,38,35);(3,4,6,5)(7,9,11,21)(8,14,16,19)(10,24,25,38)(12,22,13,28)(15,31,32,36)(17,30,18,34)(20,26,37,27)(23,35,33,29);(7,10,11,25)(8,15,16,32)(9,20,21,37)(12,27,13,26)(14,29,19,35)(17,33,18,23)(22,38,28,24)(30,36,34,31);(3,5,6,4)(8,16)(14,19)(15,32)(17,18)(23,33)(29,35)(30,34)(31,36)86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6350223);