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 | #62902 | C2 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×#117 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
2646 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 5 generators on 64 points:
(1,2,9,15)(3,16,22,50)(4,23,28,6)(5,13,30,12)(7,34,37,46)(8,14,32,25)(10,38,43,18)(11,44,33,27)(17,51,55,56)(19,49,59,52)(20,40,21,47)(24,31,61,60)(26,45,36,29)(35,42,57,41)(39,63,58,62)(48,54,64,53);(1,3)(2,10)(4,17)(5,18)(6,19)(7,35)(8,21)(9,22)(11,39)(12,40)(13,47)(14,42)(15,43)(16,34)(20,32)(23,59)(24,52)(25,41)(26,62)(27,54)(28,55)(29,56)(30,38)(31,48)(33,58)(36,63)(37,57)(44,53)(45,51)(46,50)(49,61)(60,64);(1,4)(2,11)(3,17)(5,24)(6,25)(7,36)(8,27)(9,28)(10,39)(12,45)(13,29)(14,23)(15,33)(16,48)(18,52)(19,41)(20,53)(21,54)(22,55)(26,37)(30,61)(31,34)(32,44)(35,63)(38,49)(40,51)(42,59)(43,58)(46,60)(47,56)(50,64)(57,62);(1,5,8,7)(2,12,14,46)(3,18,21,35)(4,24,27,36)(6,31,33,29)(9,30,32,37)(10,40,42,50)(11,45,23,60)(13,25,34,15)(16,43,47,41)(17,52,54,63)(19,48,58,56)(20,57,22,38)(26,28,61,44)(39,51,59,64)(49,53,62,55);(1,6,32,11)(2,4,25,44)(3,19,20,39)(5,29,37,60)(7,31,30,45)(8,33,9,23)(10,17,41,53)(12,36,34,61)(13,26,46,24)(14,27,15,28)(16,49,40,63)(18,56,57,64)(21,58,22,59)(35,48,38,51)(42,54,43,55)(47,62,50,52)Polycyclic presentation:
2^9|a^2=i,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, 6260026);