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 | C22×#2237 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C24 | C2×D8 |
| commutator, Frattini, cosocle | C2×C8 | C25 |
| prime radical | C25 | D8 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ C24×C4 ◁ C24×C8 ◁ G | |
| LECS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
1698 conjugacy classes of subgroups.
| minimal degree | μ(G) = 44 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 44 points:
(1,2);(3,4);(5,6)(7,10)(8,9)(11,12)(13,14,18,23)(15,19,26,34)(16,20,29,36)(17,30,28,22)(21,37,35,27)(24,31,40,43)(25,41,39,33)(32,44,42,38);(6,9)(10,12)(13,15,18,26)(14,19,23,34)(16,24,29,40)(17,25,28,39)(20,31,36,43)(21,32,35,42)(22,33,30,41)(27,38,37,44);(5,7,8,11)(6,10,9,12)(14,20)(16,27)(17,28)(19,31)(21,22)(23,36)(24,38)(25,39)(29,37)(30,35)(32,33)(34,43)(40,44)(41,42)116 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7530228);