Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #2184 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
| direct product | C2×#33645 |
| subgroup | G/H | |
|---|---|---|
| center | C22×C4 | C2×D8 |
| commutator, Frattini, cosocle | C2×C8 | C25 |
| socle | C23 | C22×D8 |
| prime radical | C2×#3050 | C2 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ C24×C4 ◁ C24×C8 ◁ G | |
| LECS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
1708 conjugacy classes of subgroups.
| minimal degree | μ(G) = 42 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 42 points:
(1,2);(3,4)(5,8)(6,7)(9,10)(11,12)(13,22)(14,19)(15,27)(16,31)(17,21)(18,24)(20,29)(23,38)(25,39)(26,37)(28,33)(30,41)(32,36)(34,42)(35,40);(4,7)(8,10);(3,5,6,9)(4,8,7,10)(11,13)(12,18)(14,23)(15,28)(16,32)(17,25)(19,33)(20,36)(21,35)(22,27)(24,29)(26,40)(30,42)(31,34)(37,38)(39,41);(11,14,17,16)(12,19,21,31)(13,23,25,32)(15,26,30,29)(18,33,35,34)(20,27,37,41)(22,38,39,36)(24,28,40,42)116 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7530231);