Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #2221 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C8 | #466 |
| commutator | C2×C8 | C23×C4 |
| Frattini, cosocle | C22×C8 | C24 |
| socle | C2 | C4×#466 |
| prime radical | #33617 | C2 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C8 ◁ C22×C8 ◁ C4×#131 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
637 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 64 points:
(1,2,8,14,9,16,7,15)(3,17,23,45,24,35,22,46)(4,25,28,11,29,41,32,42)(5,12,6,31,18,13,30,43)(10,33,36,53,38,20,37,21)(19,48,51,44,52,64,54,61)(26,56,27,39,47,57,58,40)(34,60,62,49,63,50,55,59);(1,3,18,33,8,23,30,53,9,24,5,20,7,22,6,21)(2,10,31,46,15,37,12,35,16,38,43,45,14,36,13,17)(4,19,47,59,28,51,58,60,29,52,26,49,32,54,27,50)(11,34,57,48,25,55,39,61,42,63,56,64,41,62,40,44);(1,4)(2,11)(3,19)(5,26)(6,27)(7,32)(8,28)(9,29)(10,34)(12,39)(13,40)(14,41)(15,25)(16,42)(17,44)(18,47)(20,49)(21,50)(22,54)(23,51)(24,52)(30,58)(31,57)(33,59)(35,61)(36,62)(37,55)(38,63)(43,56)(45,64)(46,48)(53,60);(1,5,8,6,9,18,7,30)(2,12,14,31,16,13,15,43)(3,20,23,21,24,33,22,53)(4,26,28,27,29,47,32,58)(10,35,36,46,38,17,37,45)(11,39,41,57,42,40,25,56)(19,49,51,50,52,59,54,60)(34,61,62,48,63,44,55,64)Polycyclic presentation:
2^9|a^2=g,b^2=eh,d^2=g,e^2=hi,f^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,e^a=eh,f^a=fi,h^a=hi,f^c=fi104 conjugacy classes — view the full character table.
| maximal subgroup of | #72320516, #72320529, #72320534, … |
| maximal normal subgroup of | #72320516, #72320529, #72320534, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 403089);