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 | #63068 | C2 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C4×#41 ◁ G | |
| LECS | G ▷ C42 ▷ C22 ▷ C1 | |
2518 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 64 points:
(2,8)(3,12)(5,6)(10,21)(11,18)(13,26)(14,19)(16,17)(20,23)(25,28)(27,29)(30,32)(33,34,39,36)(35,43,47,56)(37,48,50,41)(38,49,53,42)(40,52,44,46)(45,59,60,57)(51,55,63,62)(54,61,58,64);(1,2)(3,9)(4,10)(5,18)(6,8)(7,11)(12,23)(13,22)(14,24)(15,25)(16,28)(17,21)(19,20)(26,32)(27,31)(29,30)(33,35,39,47)(34,40,36,44)(37,45,50,60)(38,52,53,46)(41,54,48,58)(42,56,49,43)(51,64,63,61)(55,59,62,57);(1,3,7,14)(2,9,11,24)(4,13,15,27)(5,12,6,19)(8,20,18,23)(10,22,25,31)(16,26,17,29)(21,30,28,32)(33,36,39,34)(35,44,47,40)(37,48,50,41)(38,42,53,49)(43,52,56,46)(45,58,60,54)(51,55,63,62)(57,64,59,61);(1,4)(2,10)(3,13)(5,16)(6,17)(7,15)(8,21)(9,22)(11,25)(12,26)(14,27)(18,28)(19,29)(20,30)(23,32)(24,31)(33,37,39,50)(34,41,36,48)(35,45,47,60)(38,51,53,63)(40,54,44,58)(42,55,49,62)(43,57,56,59)(46,61,52,64);(4,15)(10,25)(13,27)(16,17)(21,28)(22,31)(26,29)(30,32)(37,50)(41,48)(45,60)(51,63)(54,58)(55,62)(57,59)(61,64)Polycyclic presentation:
2^9|a^2=i,b^2=i,c^2=h,d^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, 6259184);