Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6351885 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C29 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C22×#120 |
| commutator, Frattini, cosocle | C42 | C25 |
| prime radical | #63065 | C2 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C4×#41 ◁ G | |
| LECS | G ▷ C42 ▷ C22 ▷ C1 | |
2570 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 64 points:
(1,2,8,13)(3,14,17,19)(4,16,9,23)(5,10,20,25)(6,12,7,11)(15,28,29,31)(18,30,24,32)(21,27,22,26)(33,34,40,44)(35,45,49,56)(36,42,37,51)(38,43,39,53)(41,54,57,48)(46,61,47,60)(50,58,52,59)(55,62,64,63);(1,3)(2,9)(4,13)(5,15)(6,23)(7,16)(8,17)(10,24)(11,19)(12,14)(18,25)(20,29)(21,32)(22,30)(26,31)(27,28)(33,35)(34,41)(36,46)(37,47)(38,48)(39,54)(40,49)(42,55)(43,56)(44,57)(45,53)(50,62)(51,64)(52,63)(58,61)(59,60);(1,4,8,9)(2,3,13,17)(5,18,20,24)(6,19,7,14)(10,15,25,29)(11,16,12,23)(21,31,22,28)(26,30,27,32)(33,36,40,37)(34,42,44,51)(35,46,49,47)(38,50,39,52)(41,55,57,64)(43,58,53,59)(45,60,56,61)(48,62,54,63);(1,5,8,20)(2,10,13,25)(3,15,17,29)(4,18,9,24)(6,21,7,22)(11,26,12,27)(14,28,19,31)(16,30,23,32)(33,37,40,36)(34,42,44,51)(35,47,49,46)(38,50,39,52)(41,55,57,64)(43,59,53,58)(45,61,56,60)(48,62,54,63);(5,20)(10,25)(15,29)(18,24)(21,22)(26,27)(28,31)(30,32)(33,38,40,39)(34,43,44,53)(35,48,49,54)(36,50,37,52)(41,56,57,45)(42,58,51,59)(46,62,47,63)(55,61,64,60)Polycyclic presentation:
2^9|a^2=h,c^2=h,d^2=h,e^2=i,f^2=h,g^2=hi,b^a=bf,c^a=cg,d^a=di,f^a=fh,g^a=ghi,f^b=fh,g^b=gi,f^c=fi,g^c=ghi,e^d=eh62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6259206);