Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of D8 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C4×C16 · Schur multiplier (H2) = C22×C4 |
| direct product | C4×#1401 |
| subgroup | G/H | |
|---|---|---|
| center | C4×C16 | D4 |
| commutator | C4 | C2×C4×C16 |
| Frattini, cosocle | C22×C16 | C23 |
| socle | C22 | C2×C22⋊C16 |
| prime radical | C2×#41 | C2×C8 |
| LCS | G ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4×C16 ◁ C2×C4×C16 ◁ G | |
| LECS | G ▷ C22×C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
334 conjugacy classes of subgroups.
| minimal degree | μ(G) = 68 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 3 generators on 68 points:
(1,2,3,4);(5,6,9,15,10,16,28,41,11,17,29,42,31,44,56,64,12,18,30,43,32,45,57,65,8,14,25,38,26,39,53,62)(7,19,21,46,22,47,50,66,20,36,48,40,49,60,67,63,23,37,27,54,51,61,55,68,24,13,33,34,52,35,58,59);(5,7)(6,13)(8,24)(9,27)(10,22)(11,20)(12,23)(14,37)(15,40)(16,35)(17,19)(18,36)(21,30)(25,48)(26,52)(28,55)(29,33)(31,49)(32,51)(34,43)(38,46)(39,61)(41,63)(42,54)(44,47)(45,60)(50,57)(53,67)(56,58)(59,65)(62,66)(64,68)224 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 59085);