Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #60470 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C4 · Schur multiplier (H2) = C29×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | C22×#121 |
| commutator | C2×C4 | C24×C4 |
| Frattini, cosocle | C22×C4 | C25 |
| socle | C2 | C22×#390 |
| prime radical | C2×#3584 | C2 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C22×#41 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
4897 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 5 generators on 64 points:
(1,2)(3,8,15,28)(4,17)(5,10)(6,23)(7,11)(9,19)(12,24)(13,33,35,55)(14,26,37,51)(16,40,39,27)(18,41)(20,42)(21,31)(22,47)(25,48,50,36)(29,44)(30,45)(32,46)(34,54,56,64)(38,59,58,52)(43,60)(49,62,63,57)(53,61);(1,3)(2,8,11,28)(4,13)(5,14)(6,16)(7,15)(9,25,30,50)(10,26,31,51)(12,27,23,40)(17,33,42,55)(18,34)(19,36)(20,35)(21,37)(22,38)(24,39)(29,49,53,63)(32,52,47,59)(41,54,60,64)(43,56)(44,57)(45,48)(46,58)(61,62);(1,4,6,19,7,20,24,45)(2,9,12,17,11,30,23,42)(3,13,16,36,15,35,39,48)(5,18,22,44,21,43,46,61)(8,25,27,33,28,50,40,55)(10,29,32,41,31,53,47,60)(14,34,38,57,37,56,58,62)(26,49,52,54,51,63,59,64);(1,5)(2,10)(3,14)(4,18)(6,22)(7,21)(8,26)(9,29)(11,31)(12,32)(13,34)(15,37)(16,38)(17,41)(19,44)(20,43)(23,47)(24,46)(25,49)(27,52)(28,51)(30,53)(33,54)(35,56)(36,57)(39,58)(40,59)(42,60)(45,61)(48,62)(50,63)(55,64);(5,21)(10,31)(14,37)(18,43)(22,46)(26,51)(29,53)(32,47)(34,56)(38,58)(41,60)(44,61)(49,63)(52,59)(54,64)(57,62)Polycyclic presentation:
2^9|a^2=h,b^2=fh,c^2=g,g^2=i,b^a=bf,c^a=cg,f^a=fi,g^a=gi,f^b=fi,h^b=hi,e^d=ei116 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6279218);