Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #32947 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C210 |
| direct product | C2×#32949 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2847 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | C2×#585 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
4414 conjugacy classes of subgroups.
| minimal degree | μ(G) = 66 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 66 points:
(1,2);(3,4,9,14)(5,15,20,38)(6,11,23,33)(7,12,25,35)(8,26,16,13)(10,27,30,19)(17,36,43,58)(18,37,45,59)(21,46,48,31)(22,49,39,32)(24,50,40,34)(28,51,54,42)(29,52,55,44)(41,61,63,57)(47,56,60,65)(53,62,64,66);(3,5,16,27,9,20,8,19)(4,10,26,38,14,30,13,15)(6,17,39,51,23,43,22,42)(7,18,40,52,25,45,24,44)(11,28,49,58,33,54,32,36)(12,29,50,59,35,55,34,37)(21,41,60,66,48,63,47,62)(31,53,65,61,46,64,56,57);(3,6)(4,11)(5,17)(7,21)(8,22)(9,23)(10,28)(12,31)(13,32)(14,33)(15,36)(16,39)(18,41)(19,42)(20,43)(24,47)(25,48)(26,49)(27,51)(29,53)(30,54)(34,56)(35,46)(37,57)(38,58)(40,60)(44,62)(45,63)(50,65)(52,66)(55,64)(59,61);(3,7)(4,12)(5,18)(6,21)(8,24)(9,25)(10,29)(11,31)(13,34)(14,35)(15,37)(16,40)(17,41)(19,44)(20,45)(22,47)(23,48)(26,50)(27,52)(28,53)(30,55)(32,56)(33,46)(36,57)(38,59)(39,60)(42,62)(43,63)(49,65)(51,66)(54,64)(58,61)92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6536239);