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×#33939 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C22×D8 |
| commutator, Frattini, cosocle | C2×C8 | C25 |
| prime radical | C22×D8 | C23 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×C4×Q8 ◁ C2×C8×Q8 ◁ G | |
| LECS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
1318 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);(4,7)(8,9)(11,12,16,21)(13,17,22,32)(14,18,27,35)(15,28,26,20)(19,36,34,25)(23,33,37,29)(24,41,38,31)(30,39,42,40);(3,4,6,7)(5,9,10,8)(11,13)(12,17)(14,23)(15,24)(16,22)(18,33)(19,30)(20,31)(21,32)(25,40)(26,38)(27,37)(28,41)(29,35)(34,42)(36,39);(12,18)(13,22)(14,25)(15,26)(17,29)(19,20)(21,35)(23,39)(27,36)(28,34)(30,41)(31,42)(32,33)(37,40);(3,5,6,10)(4,8,7,9)86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7531060);