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 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C4×#41 ◁ G | |
| LECS | G ▷ C42 ▷ C22 ▷ C1 | |
2519 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 64 points:
(1,2,8,5)(3,13,18,26)(4,19,22,25)(6,11,7,12)(9,16,15,17)(10,23,20,21)(14,24,31,28)(27,30,32,29)(33,34)(35,44)(36,42)(37,40)(38,43)(39,52)(41,47)(45,60)(46,53)(48,56)(49,51)(50,57)(54,58)(55,63)(59,64)(61,62);(1,3)(2,9)(4,14)(5,15)(6,16)(7,17)(8,18)(10,24)(11,13)(12,26)(19,32)(20,28)(21,29)(22,31)(23,30)(25,27)(33,35,40,48)(34,41,37,46)(36,45,51,64)(38,47,39,53)(42,55,49,61)(43,56,52,44)(50,62,54,63)(57,60,58,59);(1,4)(2,10)(3,14)(5,20)(6,21)(7,23)(8,22)(9,24)(11,25)(12,19)(13,27)(15,28)(16,29)(17,30)(18,31)(26,32)(33,36)(34,42)(35,45)(37,49)(38,50)(39,54)(40,51)(41,55)(43,57)(44,59)(46,61)(47,62)(48,64)(52,58)(53,63)(56,60);(1,5,8,2)(3,15,18,9)(4,20,22,10)(6,11,7,12)(13,17,26,16)(14,28,31,24)(19,21,25,23)(27,30,32,29)(33,37,40,34)(35,46,48,41)(36,49,51,42)(38,43,39,52)(44,47,56,53)(45,61,64,55)(50,57,54,58)(59,62,60,63);(1,6,8,7)(2,11,5,12)(3,16,18,17)(4,21,22,23)(9,13,15,26)(10,25,20,19)(14,29,31,30)(24,27,28,32)(33,38,40,39)(34,43,37,52)(35,47,48,53)(36,50,51,54)(41,56,46,44)(42,57,49,58)(45,62,64,63)(55,60,61,59)Polycyclic presentation:
2^9|a^2=hi,b^2=i,d^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, 6259136);