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 | |
2514 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 64 points:
(1,2,8,13)(3,14,9,23)(4,17,19,16)(5,10,20,25)(6,12,7,11)(15,28,24,32)(18,30,31,29)(21,27,22,26)(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);(1,3,8,9)(2,4,13,19)(5,15,20,24)(6,17,7,16)(10,18,25,31)(11,23,12,14)(21,30,22,29)(26,32,27,28)(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)(3,13)(5,18)(6,14)(7,23)(8,19)(10,24)(11,17)(12,16)(15,25)(20,31)(21,28)(22,32)(26,30)(27,29)(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);(1,5,8,20)(2,10,13,25)(3,15,9,24)(4,18,19,31)(6,21,7,22)(11,26,12,27)(14,28,23,32)(16,29,17,30)(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);(5,20)(10,25)(15,24)(18,31)(21,22)(26,27)(28,32)(29,30)(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=h,b^2=h,d^2=h,e^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, 6259204);