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 | #63053 | C2 |
| 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)(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)(33,34,39,43)(35,44,47,55)(36,48,50,41)(37,42,38,52)(40,53,56,46)(45,60,63,59)(49,58,51,57)(54,62,64,61);(1,3,7,17)(2,8,12,23)(4,14,20,30)(5,15,6,16)(9,22,25,32)(10,13,11,26)(18,31,24,27)(19,28,21,29)(33,35,39,47)(34,40,43,56)(36,45,50,63)(37,46,38,53)(41,54,48,64)(42,55,52,44)(49,61,51,62)(57,59,58,60);(1,4)(2,9)(3,14)(5,19)(6,21)(7,20)(8,22)(10,24)(11,18)(12,25)(13,27)(15,28)(16,29)(17,30)(23,32)(26,31)(33,34)(35,40)(36,41)(37,42)(38,52)(39,43)(44,53)(45,54)(46,55)(47,56)(48,50)(49,57)(51,58)(59,61)(60,62)(63,64);(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)(33,36,39,50)(34,41,43,48)(35,45,47,63)(37,51,38,49)(40,54,56,64)(42,58,52,57)(44,59,55,60)(46,62,53,61);(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)(33,37,39,38)(34,42,43,52)(35,46,47,53)(36,49,50,51)(40,55,56,44)(41,57,48,58)(45,61,63,62)(54,59,64,60)Polycyclic presentation:
2^9|a^2=i,b^2=h,d^2=i,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, 6259161);