Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #509 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C42 · Schur multiplier (H2) = C25×C4 |
| direct product | C4×#2143 |
| subgroup | G/H | |
|---|---|---|
| center | C42 | C2×D8 |
| commutator | C8 | C22×C42 |
| Frattini, cosocle | C22×C8 | C24 |
| socle | C22 | C22×#102 |
| prime radical | C22×#135 | C22 |
| LCS | G ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C42 ◁ C22×C42 ◁ C22×C4×C8 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
1184 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 36 points:
(1,2,3,4);(5,6,9,14,11,15,10,25)(7,16,19,33,17,32,26,30)(8,20,22,31,24,35,23,13)(12,21,28,34,27,36,29,18);(5,7)(6,12)(8,21)(9,19)(10,26)(11,17)(13,30)(14,28)(15,27)(16,20)(18,23)(22,34)(24,36)(25,29)(31,33)(32,35);(7,17)(12,27)(16,32)(18,34)(19,26)(21,36)(28,29)(30,33)128 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 400253);