Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #502149 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #25108 |
| commutator, Frattini, cosocle | C22×C8 | C24 |
| prime radical | #24297 | C2 |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C22×C4 ◁ #2955 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
724 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 64 points:
(1,2,7,15,6,34,8,16,10,17,36,51,9,14,37,50)(3,18,22,52,35,44,38,61,24,43,60,42,21,11,23,41)(4,25,28,46,31,13,29,48,5,30,32,47,27,49,33,12)(19,45,56,39,55,62,20,40,58,63,59,54,57,64,26,53);(1,3,7,22,6,35,8,38,10,24,36,60,9,21,37,23)(2,11,15,41,34,18,16,52,17,44,51,61,14,43,50,42)(4,26,28,19,31,56,29,55,5,20,32,58,27,59,33,57)(12,45,25,39,46,62,13,40,48,63,30,54,47,64,49,53);(1,4,10,5)(2,12,17,48)(3,19,24,58)(6,27,9,31)(7,33,36,29)(8,32,37,28)(11,39,44,54)(13,15,49,51)(14,46,34,47)(16,30,50,25)(18,53,43,40)(20,22,26,60)(21,55,35,57)(23,56,38,59)(41,45,61,63)(42,62,52,64);(1,5,10,4)(2,13,17,49)(3,20,24,26)(6,27,9,31)(7,32,36,28)(8,33,37,29)(11,40,44,53)(12,15,48,51)(14,25,34,30)(16,47,50,46)(18,54,43,39)(19,22,58,60)(21,56,35,59)(23,55,38,57)(41,63,61,45)(42,62,52,64)Polycyclic presentation:
2^9|a^2=f,b^2=f,c^2=i,d^2=i,e^2=i,f^2=hi,g^2=h,h^2=i,b^a=be,c^a=cf,d^a=de,e^a=ei,c^b=cg,e^b=ei,g^b=gi,f^c=fh,g^c=ghi,h^c=hi,e^d=ei62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 409348);