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 | #63055 | C2 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C4×#41 ◁ G | |
| LECS | G ▷ C42 ▷ C22 ▷ C1 | |
2632 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 64 points:
(1,2,8,14)(3,15,18,32)(4,19,21,27)(5,11,24,31)(6,12,7,13)(9,16,26,17)(10,25,28,20)(22,29,23,30)(33,34,38,42)(35,43,46,56)(36,47,48,40)(37,50,52,41)(39,51,54,45)(44,58,59,57)(49,55,63,62)(53,60,61,64);(1,3)(2,9)(4,5)(6,16)(7,17)(8,18)(10,11)(12,15)(13,32)(14,26)(19,30)(20,22)(21,24)(23,25)(27,29)(28,31)(33,35,38,46)(34,39,42,54)(36,44,48,59)(37,51,52,45)(40,53,47,61)(41,56,50,43)(49,64,63,60)(55,58,62,57);(1,4)(2,10)(3,5)(6,20)(7,25)(8,21)(9,11)(12,27)(13,19)(14,28)(15,29)(16,22)(17,23)(18,24)(26,31)(30,32)(37,52)(41,50)(43,56)(45,51)(49,63)(55,62)(57,58)(60,64);(1,5)(2,11)(3,4)(6,23)(7,22)(8,24)(9,10)(12,30)(13,29)(14,31)(15,19)(16,25)(17,20)(18,21)(26,28)(27,32)(33,36)(34,40)(35,44)(37,49)(38,48)(39,53)(41,55)(42,47)(43,57)(45,60)(46,59)(50,62)(51,64)(52,63)(54,61)(56,58);(1,6,8,7)(2,12,14,13)(3,16,18,17)(4,20,21,25)(5,22,24,23)(9,15,26,32)(10,27,28,19)(11,29,31,30)(36,48)(40,47)(44,59)(49,63)(53,61)(55,62)(57,58)(60,64)Polycyclic presentation:
2^9|a^2=h,b^2=i,e^2=hi,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, 6259242);