Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6352701 |
| 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 | C22×C4 | C25 |
| prime radical | #62934 | C2 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×#117 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
2608 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 5 generators on 64 points:
(1,2,8,13)(3,15,21,38)(4,6,12,33)(5,27,29,58)(7,35,30,11)(9,14,22,24)(10,36,41,19)(16,46,39,61)(17,40,51,49)(18,44,54,48)(20,45,47,60)(23,32,25,59)(26,34,42,56)(28,57,31,37)(43,63,52,50)(53,62,64,55);(1,3)(2,10)(4,16)(5,17)(6,18)(7,36)(8,21)(9,20)(11,38)(12,39)(13,41)(14,40)(15,35)(19,30)(22,47)(23,48)(24,49)(25,44)(26,50)(27,60)(28,43)(29,51)(31,52)(32,61)(33,54)(34,53)(37,62)(42,63)(45,58)(46,59)(55,57)(56,64);(1,4,22,42)(2,6,24,56)(3,16,47,63)(5,23,7,37)(8,12,9,26)(10,18,49,64)(11,31,58,59)(13,33,14,34)(15,43,60,61)(17,48,36,62)(19,55,51,44)(20,50,21,39)(25,30,57,29)(27,32,35,28)(38,52,45,46)(40,53,41,54);(1,5,8,29)(2,11,13,35)(3,17,21,51)(4,23,12,25)(6,31,33,28)(7,9,30,22)(10,38,41,15)(14,27,24,58)(16,48,39,44)(18,52,54,43)(19,47,36,20)(26,57,42,37)(32,56,59,34)(40,60,49,45)(46,53,61,64)(50,55,63,62);(1,6,9,34)(2,12,14,42)(3,18,20,53)(4,24,26,13)(5,28,30,59)(7,32,29,31)(8,33,22,56)(10,39,40,63)(11,23,27,57)(15,44,45,62)(16,49,50,41)(17,43,19,46)(21,54,47,64)(25,58,37,35)(36,61,51,52)(38,48,60,55)Polycyclic presentation:
2^9|a^2=h,c^2=hi,d^2=h,e^2=i,f^2=h,b^a=bf,c^a=cg,d^a=di,f^a=fh,f^b=fh,g^b=gi,f^c=fi,e^d=eh65 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6260014);