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 | |
2500 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)(3,13)(4,18)(5,10)(6,11)(7,12)(8,16)(9,19)(14,22)(15,23)(17,26)(20,24)(21,25)(27,28)(29,31)(30,32)(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)(2,8)(4,14)(5,15)(6,16)(7,17)(9,22)(10,13)(11,26)(12,23)(18,31)(19,28)(20,30)(21,29)(24,27)(25,32)(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);(1,4,7,20)(2,9,12,25)(3,14,17,30)(5,19,6,21)(8,22,23,32)(10,24,11,18)(13,27,26,31)(15,28,16,29)(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);(1,2)(3,8)(4,9)(5,11)(6,10)(7,12)(13,16)(14,22)(15,26)(17,23)(18,19)(20,25)(21,24)(27,29)(28,31)(30,32)(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);(1,5,7,6)(2,10,12,11)(3,15,17,16)(4,19,20,21)(8,13,23,26)(9,24,25,18)(14,28,30,29)(22,27,32,31)(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|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, 6259120);