Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #33830 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
| direct product | C2×#33884 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C22×D8 |
| commutator, Frattini, cosocle | C2×C8 | C25 |
| prime radical | C2×#3287 | C2 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×C4×D4 ◁ C2×C8×D4 ◁ G | |
| LECS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
1568 conjugacy classes of subgroups.
| minimal degree | μ(G) = 42 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 42 points:
(1,2);(3,4,6,8)(5,9,10,7)(11,12,17,21)(13,22,23,39)(14,19,16,31)(15,27,30,40)(18,33,34,25)(20,26,38,29)(24,41,32,37)(28,35,42,36);(3,5)(4,7)(6,10)(8,9)(13,23)(18,34)(22,39)(24,32)(25,33)(28,42)(35,36)(37,41);(11,13,17,23)(12,18,21,34)(14,24,16,32)(15,28,30,42)(19,35,31,36)(20,37,38,41)(22,40,39,27)(25,26,33,29);(5,10)(7,9)(11,14,17,16)(12,19,21,31)(13,24,23,32)(15,26,30,29)(18,35,34,36)(20,27,38,40)(22,41,39,37)(25,28,33,42)86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7530893);