Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6447052 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | C2×#2420 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | C2×#575 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1769 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)(3,13,16,8)(4,18,21,46)(5,10,25,37)(6,26,27,11)(9,31,34,22)(14,39,19,47)(15,35,23,29)(17,30,40,38)(20,45,50,62)(24,53,43,36)(28,42,32,49)(33,56,58,51)(41,60,48,63)(44,55,52,59)(54,61,57,64);(1,3,6,17,7,16,27,40)(2,8,11,30,12,13,26,38)(4,19,22,49,21,14,31,42)(5,23,24,52,25,15,43,44)(9,32,18,39,34,28,46,47)(10,35,36,59,37,29,53,55)(20,48,51,64,50,41,56,61)(33,57,45,60,58,54,62,63);(1,4,6,22,7,21,27,31)(2,9,11,18,12,34,26,46)(3,14,40,49,16,19,17,42)(5,20,24,51,25,50,43,56)(8,28,38,39,13,32,30,47)(10,33,36,45,37,58,53,62)(15,41,52,64,23,48,44,61)(29,54,59,60,35,57,55,63);(1,5)(2,10)(3,15,16,23)(4,20)(6,24)(7,25)(8,29,13,35)(9,33)(11,36)(12,37)(14,41,19,48)(17,44,40,52)(18,45)(21,50)(22,51)(26,53)(27,43)(28,54,32,57)(30,55,38,59)(31,56)(34,58)(39,60,47,63)(42,61,49,64)(46,62);(4,21)(9,34)(14,19)(18,46)(20,50)(22,31)(28,32)(33,58)(39,47)(41,48)(42,49)(45,62)(51,56)(54,57)(60,63)(61,64)Polycyclic presentation:
2^9|a^2=i,b^2=g,c^2=fg,d^2=f,g^2=i,b^a=bf,c^a=cg,g^a=gi,c^b=ch,d^b=dh,f^b=fi,e^c=ei,h^c=hi,h^d=hi77 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6354249);