Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #493411 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | C2×#1781 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C2×#41 | C24 |
| prime radical | #62859 | C2 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C22×C4 ◁ C2×#585 ◁ G | |
| LECS | G ▷ C2×#41 ▷ C22 ▷ C2 ▷ C1 | |
3249 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 64 points:
(1,2)(3,15)(4,20,23,38)(5,11)(6,26,28,12)(7,13)(8,14)(9,18,32,48)(10,21,34,29)(16,42,46,61)(17,40)(19,41)(22,52)(24,54)(25,37,27,36)(30,58)(31,47,35,55)(33,51)(39,45,62,43)(44,59)(49,64)(50,53)(56,57)(60,63);(1,3,8,19)(2,9,14,32)(4,16,23,46)(5,24,7,17)(6,18,28,48)(10,30,34,60)(11,35,13,31)(12,15,26,41)(20,49,38,59)(21,45,29,43)(22,44,53,64)(25,55,27,47)(33,39,56,62)(36,54,37,40)(42,52,61,50)(51,58,57,63);(1,4)(2,10)(3,16)(5,21)(6,22)(7,29)(8,23)(9,30)(11,20)(12,33)(13,38)(14,34)(15,39)(17,43)(18,44)(19,46)(24,45)(25,51)(26,56)(27,57)(28,53)(31,59)(32,60)(35,49)(36,50)(37,52)(40,61)(41,62)(42,54)(47,63)(48,64)(55,58);(1,5,8,7)(2,11,14,13)(3,17,19,24)(4,21,23,29)(6,25,28,27)(9,31,32,35)(10,20,34,38)(12,36,26,37)(15,40,41,54)(16,43,46,45)(18,47,48,55)(22,51,53,57)(30,59,60,49)(33,50,56,52)(39,61,62,42)(44,63,64,58)Polycyclic presentation:
2^9|a^2=g,b^2=i,d^2=i,f^2=i,b^a=be,c^a=cf,e^a=eh,h^a=hi,d^b=di,g^b=gh,f^c=fi,g^c=gi,g^e=gi65 conjugacy classes — view the full character table.
| maximal subgroup of | #72302990, #72303033, #72303035, … |
| maximal normal subgroup of | #72302990, #72303033, #72303035, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 400610);