Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #13033 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #1879 |
| commutator | C2×C8 | C23×C4 |
| Frattini, cosocle | C22×C8 | C24 |
| socle | C2 | C2×#1879 |
| prime radical | C2×#3414 | C2 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C23×C4 ◁ C2×#405 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
1299 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 64 points:
(1,2,7,13,8,15,28,43,9,16,6,14,5,12,29,42)(3,17,21,37,19,44,32,38,22,51,20,48,34,46,30,10)(4,23,26,49,24,45,31,11,27,40,25,39,35,47,33,41)(18,36,54,60,55,61,57,63,56,62,53,64,52,59,58,50);(1,3)(2,10,16,38)(4,18)(5,19)(6,30)(7,32)(8,34)(9,22)(11,36,41,62)(12,37,15,48)(13,44,14,46)(17,42,51,43)(20,29)(21,28)(23,50,40,63)(24,52)(25,58)(26,57)(27,56)(31,54)(33,53)(35,55)(39,59,49,61)(45,64,47,60);(1,4)(2,11)(3,18)(5,24)(6,31)(7,33)(8,35)(9,27)(10,36)(12,39)(13,45)(14,47)(15,49)(16,41)(17,50)(19,52)(20,57)(21,58)(22,56)(23,43)(25,28)(26,29)(30,54)(32,53)(34,55)(37,59)(38,62)(40,42)(44,64)(46,60)(48,61)(51,63);(1,5,9,8)(2,12,16,15)(3,19,22,34)(4,24,27,35)(6,28,7,29)(10,37,38,48)(11,39,41,49)(13,42,14,43)(17,44,51,46)(18,52,56,55)(20,30,21,32)(23,45,40,47)(25,33,26,31)(36,59,62,61)(50,64,63,60)(53,57,54,58)Polycyclic presentation:
2^9|a^2=g,b^2=egh,d^2=i,e^2=h,f^2=h,g^2=h,h^2=i,b^a=be,c^a=cf,e^a=ei,e^b=eh,f^b=fhi,g^b=gh,h^b=hi,e^c=ehi,f^c=fhi,g^c=ghi,h^c=hi80 conjugacy classes — view the full character table.
| maximal subgroup of | #72280165, #72280389, #72281086, … |
| maximal normal subgroup of | #72280165, #72280389, #72281086, … |
| maximal quotient of | C2×#494553, #72310139, #72342558 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 401749);