Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6353061 |
| 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 | #62906 | C2 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C4×D4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
2683 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 5 generators on 64 points:
(1,2,9,14)(3,15,19,46)(4,5,26,30)(6,12,32,43)(7,33,36,57)(8,13,21,22)(10,37,40,18)(11,42,25,29)(16,47,52,44)(17,50,55,51)(20,39,48,63)(23,27,58,28)(24,31,35,60)(34,41,62,49)(38,54,53,61)(45,64,59,56);(1,3)(2,10)(4,16)(5,17)(6,18)(7,34)(8,20)(9,19)(11,38)(12,39)(13,41)(14,40)(15,33)(21,48)(22,49)(23,50)(24,59)(25,53)(26,52)(27,54)(28,61)(29,56)(30,55)(31,47)(32,37)(35,45)(36,62)(42,64)(43,63)(44,60)(46,57)(51,58);(1,4,21,11)(2,5,22,42)(3,16,48,38)(6,23,36,24)(7,35,32,58)(8,25,9,26)(10,17,49,64)(12,27,57,31)(13,29,14,30)(15,44,63,61)(18,50,62,59)(19,52,20,53)(28,33,60,43)(34,45,37,51)(39,54,46,47)(40,55,41,56);(1,5)(2,11)(3,17)(4,22)(6,31)(7,28)(8,29)(9,30)(10,38)(12,23)(13,26)(14,25)(15,45)(16,49)(18,47)(19,55)(20,56)(21,42)(24,57)(27,36)(32,60)(33,35)(34,61)(37,44)(39,50)(40,53)(41,52)(43,58)(46,59)(48,64)(51,63)(54,62);(1,6,8,7)(2,12,13,33)(3,18,20,34)(4,23,25,35)(5,27,29,60)(9,32,21,36)(10,39,41,15)(11,24,26,58)(14,43,22,57)(16,50,53,45)(17,54,56,44)(19,37,48,62)(28,42,31,30)(38,59,52,51)(40,63,49,46)(47,55,61,64)Polycyclic presentation:
2^9|a^2=i,c^2=hi,e^2=h,f^2=h,b^a=bf,c^a=cg,d^a=dh,f^a=fh,f^b=fh,g^b=gi,f^c=fi,e^d=ehi65 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6260429);