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 | #62911 | C2 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×#117 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
2609 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 5 generators on 64 points:
(1,2,9,16)(3,17,21,50)(4,11,25,30)(5,14,27,12)(6,13,31,46)(7,32,35,44)(8,15,29,43)(10,36,40,19)(18,51,54,48)(20,49,57,60)(22,47,56,38)(23,26,45,42)(24,59,34,28)(33,41,61,63)(37,52,58,53)(39,62,64,55);(1,3)(2,10)(4,18)(5,19)(6,20)(7,33)(8,22)(9,21)(11,37)(12,38)(13,39)(14,47)(15,41)(16,40)(17,32)(23,52)(24,49)(25,54)(26,55)(27,36)(28,48)(29,56)(30,58)(31,57)(34,60)(35,61)(42,62)(43,63)(44,50)(45,53)(46,64)(51,59);(1,4,8,13)(2,11,15,31)(3,18,22,39)(5,23,7,34)(6,16,30,43)(9,25,29,46)(10,37,41,57)(12,42,44,59)(14,26,32,28)(17,48,47,55)(19,52,33,60)(20,40,58,63)(21,54,56,64)(24,27,45,35)(36,53,61,49)(38,62,50,51);(1,5,8,7)(2,12,15,44)(3,19,22,33)(4,23,13,34)(6,28,30,26)(9,27,29,35)(10,38,41,50)(11,42,31,59)(14,43,32,16)(17,40,47,63)(18,52,39,60)(20,48,58,55)(21,36,56,61)(24,25,45,46)(37,62,57,51)(49,54,53,64);(1,6,29,11)(2,13,43,25)(3,20,56,37)(4,16,46,15)(5,26,35,59)(7,28,27,42)(8,30,9,31)(10,39,63,54)(12,23,32,24)(14,45,44,34)(17,49,38,52)(18,40,64,41)(19,55,61,51)(21,57,22,58)(33,48,36,62)(47,53,50,60)Polycyclic presentation:
2^9|a^2=i,c^2=h,d^2=h,e^2=hi,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, 6260046);