Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7157001 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #44936 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1723 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 5 generators on 64 points:
(1,2,8,14,9,4,7,13)(3,15,34,10,20,16,19,21)(5,11,35,40,28,22,27,25)(6,29,32,23,33,42,31,12)(17,38,47,36,48,44,24,45)(18,37,58,41,49,50,30,43)(26,54,59,51,57,53,56,39)(46,60,64,62,55,63,52,61);(1,3)(2,10)(4,21)(5,24)(6,30)(7,34)(8,19)(9,20)(11,38)(12,41)(13,16)(14,15)(17,27)(18,31)(22,44)(23,43)(25,36)(26,55)(28,47)(29,37)(32,49)(33,58)(35,48)(39,60)(40,45)(42,50)(46,57)(51,63)(52,56)(53,62)(54,61)(59,64);(1,4)(2,9)(3,16,20,15)(5,25,28,40)(6,23)(7,13)(8,14)(10,19,21,34)(11,35,22,27)(12,33)(17,45)(18,41,49,43)(24,38)(26,53,57,54)(29,32)(30,50,58,37)(31,42)(36,48)(39,59,51,56)(44,47)(46,63)(52,61)(55,60)(62,64);(1,5,7,27,9,28,8,35)(2,11,13,25,4,22,14,40)(3,17,19,24,20,48,34,47)(6,26,31,56,33,57,32,59)(10,36,15,38,21,45,16,44)(12,39,42,53,23,51,29,54)(18,46,30,52,49,55,58,64)(37,60,43,61,50,63,41,62);(1,6,8,32,9,33,7,31)(2,12,13,42,4,23,14,29)(3,18,34,58,20,49,19,30)(5,26,35,59,28,57,27,56)(10,37,15,43,21,50,16,41)(11,39,25,53,22,51,40,54)(17,46,47,64,48,55,24,52)(36,60,38,61,45,63,44,62)Polycyclic presentation:
2^9|a^2=fi,c^2=fg,d^2=f,e^2=hi,f^2=i,g^2=i,h^2=i,b^a=bf,c^a=cg,e^a=eh,h^a=hi,c^b=ch,d^b=df,e^b=eh,f^b=fi,h^b=hi,d^c=dh,f^c=fi,g^c=gi,g^d=gi62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7064200);