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 | #62933 | C2 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×#117 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
2675 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 5 generators on 64 points:
(1,2,9,30)(3,16,31,34)(4,11,6,28)(5,13,32,41)(7,15,8,12)(10,33,55,19)(14,23,39,24)(17,35,29,37)(18,46,56,58)(20,45,21,44)(22,38,26,53)(25,43,27,40)(36,57,60,49)(42,51,62,52)(47,59,54,61)(48,63,50,64);(1,3)(2,10)(4,17)(5,18)(6,29)(7,21)(8,20)(9,31)(11,34)(12,35)(13,36)(14,44)(15,37)(16,28)(19,24)(22,47)(23,33)(25,48)(26,54)(27,50)(30,55)(32,56)(38,58)(39,45)(40,59)(41,60)(42,63)(43,61)(46,53)(49,52)(51,57)(62,64);(1,4,9,6)(2,11,30,28)(3,17,31,29)(5,22,32,26)(7,23,8,24)(10,34,55,16)(12,14,15,39)(13,38,41,53)(18,47,56,54)(19,21,33,20)(25,51,27,52)(35,44,37,45)(36,58,60,46)(40,42,43,62)(48,57,50,49)(59,63,61,64);(2,12)(5,25)(10,35)(11,14)(13,41)(15,30)(16,45)(18,48)(22,51)(26,52)(27,32)(28,39)(34,44)(36,60)(37,55)(38,53)(40,43)(42,62)(46,58)(47,57)(49,54)(50,56)(59,61)(63,64);(1,5,8,27)(2,13,12,40)(3,18,20,50)(4,22,24,52)(6,26,23,51)(7,25,9,32)(10,36,35,59)(11,38,14,42)(15,43,30,41)(16,46,45,64)(17,47,19,49)(21,48,31,56)(28,53,39,62)(29,54,33,57)(34,58,44,63)(37,61,55,60)Polycyclic presentation:
2^9|a^2=hi,c^2=hi,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, 6260133);