Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #33856 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C29 |
| direct product | C2×#33860 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | D4×D8 |
| commutator, Frattini, cosocle | C2×C8 | C25 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ C2×#443 ◁ G | |
| LECS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
1886 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);(3,4,11,17)(5,18,23,53)(6,24,13,39)(7,28,31,27)(8,32,37,48)(9,16,10,15)(12,40,44,21)(14,35,49,36)(19,20,25,58)(22,60,38,46)(26,56,45,30)(29,63,41,42)(33,50,65,43)(34,47,66,64)(51,52,57,55)(54,59,62,61);(3,5,11,23)(4,12,17,44)(6,25,13,19)(7,29,31,41)(8,33,37,65)(9,22,10,38)(14,46,49,60)(15,43,16,50)(18,48,53,32)(20,56,58,30)(21,35,40,36)(24,61,39,59)(26,63,45,42)(27,52,28,55)(34,57,66,51)(47,54,64,62);(3,6,11,13)(4,7,17,31)(5,19,23,25)(8,26,37,45)(9,28,10,27)(12,41,44,29)(14,30,49,56)(15,39,16,24)(18,51,53,57)(20,46,58,60)(21,54,40,62)(22,52,38,55)(32,34,48,66)(33,42,65,63)(35,47,36,64)(43,61,50,59);(3,7)(4,13)(5,20)(6,17)(8,34)(9,24)(10,39)(11,31)(12,42)(14,47)(15,28)(16,27)(18,52)(19,46)(21,59)(22,57)(23,58)(25,60)(26,48)(29,33)(30,36)(32,45)(35,56)(37,66)(38,51)(40,61)(41,65)(43,54)(44,63)(49,64)(50,62)(53,55)86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7530856);