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 | #62929 | C2 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×#117 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
2640 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 5 generators on 64 points:
(1,2,8,15)(3,17,22,48)(4,24,27,11)(5,14,7,32)(6,13,31,47)(9,16,36,44)(10,33,41,19)(12,35,45,29)(18,52,55,64)(20,50,60,63)(21,38,23,51)(25,28,34,30)(26,43,46,61)(37,53,59,62)(39,58,54,49)(40,57,42,56);(1,3,8,22)(2,10,15,41)(4,18,27,55)(5,19,7,33)(6,20,31,60)(9,23,36,21)(11,37,24,59)(12,38,45,51)(13,39,47,54)(14,48,32,17)(16,42,44,40)(25,50,34,63)(26,62,46,53)(28,49,30,58)(29,57,35,56)(43,52,61,64);(1,4,9,13)(2,11,16,6)(3,18,23,39)(5,25,29,46)(7,34,35,26)(8,27,36,47)(10,37,42,20)(12,43,14,30)(15,24,44,31)(17,49,51,64)(19,50,57,53)(21,54,22,55)(28,45,61,32)(33,63,56,62)(38,52,48,58)(40,60,41,59);(1,5,8,7)(2,12,15,45)(3,19,22,33)(4,25,27,34)(6,30,31,28)(9,29,36,35)(10,38,41,51)(11,43,24,61)(13,46,47,26)(14,44,32,16)(17,42,48,40)(18,50,55,63)(20,58,60,49)(21,56,23,57)(37,52,59,64)(39,53,54,62);(1,6,9,11)(2,13,16,4)(3,20,23,37)(5,28,29,61)(7,30,35,43)(8,31,36,24)(10,39,42,18)(12,26,14,34)(15,47,44,27)(17,50,51,53)(19,49,57,64)(21,59,22,60)(25,45,46,32)(33,58,56,52)(38,62,48,63)(40,55,41,54)Polycyclic presentation:
2^9|a^2=h,b^2=h,c^2=i,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, 6260004);