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 |
| prime radical | #61106 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1729 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 5 generators on 64 points:
(1,2,7,14,9,16,8,15)(3,17,5,12,20,35,28,10)(4,21,24,11,26,38,25,39)(6,29,30,13,32,43,31,44)(18,33,48,36,49,40,22,45)(19,34,27,46,50,41,57,42)(23,37,54,58,56,62,55,51)(47,60,64,61,53,63,52,59);(1,3,9,20)(2,10,16,12)(4,22,26,48)(5,7,28,8)(6,27,32,57)(11,36,39,45)(13,42,44,46)(14,35,15,17)(18,24,49,25)(19,31,50,30)(21,40,38,33)(23,53,56,47)(29,34,43,41)(37,61,62,59)(51,63,58,60)(52,54,64,55);(1,4)(2,11)(3,18,20,49)(5,22,28,48)(6,23)(7,25)(8,24)(9,26)(10,33,12,40)(13,37)(14,21)(15,38)(16,39)(17,45,35,36)(19,47,50,53)(27,52,57,64)(29,58)(30,55)(31,54)(32,56)(34,59,41,61)(42,60,46,63)(43,51)(44,62);(1,5,7,20,9,28,8,3)(2,12,14,35,16,10,15,17)(4,18,24,48,26,49,25,22)(6,27,30,50,32,57,31,19)(11,36,38,40,39,45,21,33)(13,41,43,42,44,34,29,46)(23,47,54,64,56,53,55,52)(37,60,58,61,62,63,51,59);(1,6,8,31,9,32,7,30)(2,13,14,43,16,44,15,29)(3,19,28,57,20,50,5,27)(4,23,24,54,26,56,25,55)(10,34,17,46,12,41,35,42)(11,37,21,51,39,62,38,58)(18,47,48,64,49,53,22,52)(33,59,45,63,40,61,36,60)Polycyclic presentation:
2^9|a^2=f,b^2=i,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, 7064201);