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 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | C22×#121 |
| commutator | C2×C4 | C24×C4 |
| Frattini, cosocle | C22×C4 | C25 |
| socle | C2 | C2×C4×#121 |
| prime radical | C2×#3067 | C2 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C22×M16 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2581 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 5 generators on 64 points:
(1,2,7,12,8,13,6,28)(3,14,18,31,19,39,17,9)(4,20,23,10,24,33,22,34)(5,11,27,36,25,35,26,52)(15,29,42,37,43,54,41,47)(16,38,46,55,44,59,45,30)(21,48,51,32,49,57,50,56)(40,53,62,58,60,64,61,63);(1,3,6,17,8,19,7,18)(2,9,12,14,13,31,28,39)(4,15,22,41,24,43,23,42)(5,16,26,45,25,44,27,46)(10,29,33,37,34,54,20,47)(11,30,36,38,35,55,52,59)(21,40,50,61,49,60,51,62)(32,53,57,58,56,64,48,63);(1,4)(2,10)(3,15)(5,21)(6,22)(7,23)(8,24)(9,29)(11,32)(12,33)(13,34)(14,37)(16,40)(17,41)(18,42)(19,43)(20,28)(25,49)(26,50)(27,51)(30,53)(31,54)(35,56)(36,57)(38,58)(39,47)(44,60)(45,61)(46,62)(48,52)(55,64)(59,63);(1,5,7,27,8,25,6,26)(2,11,12,36,13,35,28,52)(3,16,18,46,19,44,17,45)(4,21,23,51,24,49,22,50)(9,30,14,38,31,55,39,59)(10,32,33,57,34,56,20,48)(15,40,42,62,43,60,41,61)(29,53,37,58,54,64,47,63);(5,25)(11,35)(16,44)(21,49)(26,27)(30,55)(32,56)(36,52)(38,59)(40,60)(45,46)(48,57)(50,51)(53,64)(58,63)(61,62)Polycyclic presentation:
2^9|a^2=h,b^2=f,d^2=h,f^2=i,g^2=i,h^2=i,b^a=bf,c^a=cg,f^a=fi,g^a=gi,g^c=gi,e^d=ei116 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6280175);