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 | #63067 | C2 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C4×#41 ◁ G | |
| LECS | G ▷ C42 ▷ C22 ▷ C1 | |
2483 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 64 points:
(1,2,8,13)(3,14,18,20)(4,17,9,16)(5,10,23,27)(6,11,7,12)(15,28,30,32)(19,29,24,31)(21,25,22,26)(33,34)(35,44)(36,40)(37,49)(38,43)(39,50)(41,47)(42,52)(45,54)(46,61)(48,56)(51,58)(53,57)(55,62)(59,64)(60,63);(1,3)(2,9)(4,13)(5,15)(6,16)(7,17)(8,18)(10,24)(11,14)(12,20)(19,27)(21,29)(22,31)(23,30)(25,32)(26,28)(33,35,40,48)(34,41,36,45)(37,46,52,64)(38,47,39,54)(42,55,49,60)(43,56,50,44)(51,62,53,63)(57,59,58,61);(1,4,8,9)(2,3,13,18)(5,19,23,24)(6,20,7,14)(10,15,27,30)(11,16,12,17)(21,28,22,32)(25,29,26,31)(33,36,40,34)(35,45,48,41)(37,49,52,42)(38,50,39,43)(44,54,56,47)(46,60,64,55)(51,58,53,57)(59,62,61,63);(1,5)(2,10)(3,15)(4,19)(6,22)(7,21)(8,23)(9,24)(11,26)(12,25)(13,27)(14,28)(16,31)(17,29)(18,30)(20,32)(33,37)(34,42)(35,46)(36,49)(38,53)(39,51)(40,52)(41,55)(43,58)(44,59)(45,60)(47,63)(48,64)(50,57)(54,62)(56,61);(1,6,8,7)(2,11,13,12)(3,16,18,17)(4,20,9,14)(5,21,23,22)(10,25,27,26)(15,29,30,31)(19,28,24,32)(33,38,40,39)(34,43,36,50)(35,47,48,54)(37,51,52,53)(41,56,45,44)(42,57,49,58)(46,62,64,63)(55,59,60,61)Polycyclic presentation:
2^9|a^2=hi,b^2=i,c^2=h,e^2=h,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, 6259134);