Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #12750 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #1786 |
| commutator | C2×C8 | C23×C4 |
| Frattini, cosocle | C22×C8 | C24 |
| socle | C2 | C2×#1786 |
| prime radical | C2×#3584 | C2 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C23×C4 ◁ C22×#131 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
3129 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 64 points:
(1,2,5,10,7,13,26,12)(3,14,18,38,19,42,16,41)(4,20,23,36,24,34,21,9)(6,27,25,11,29,37,48,39)(8,28,32,17,33,49,31,50)(15,43,44,40,47,61,46,62)(22,54,56,60,57,35,53,52)(30,45,58,55,59,64,51,63);(1,3)(2,8)(4,15)(5,16)(6,28)(7,19)(9,30)(10,31)(11,38)(12,32)(13,33)(14,37)(17,48)(18,26)(20,51)(21,44)(22,55)(23,46)(24,47)(25,50)(27,42)(29,49)(34,58)(35,43)(36,59)(39,41)(40,60)(45,56)(52,62)(53,64)(54,61)(57,63);(1,4)(2,9)(3,15)(5,21)(6,22)(7,24)(8,30)(10,34)(11,35)(12,20)(13,36)(14,40)(16,44)(17,45)(18,46)(19,47)(23,26)(25,53)(27,52)(28,55)(29,57)(31,58)(32,51)(33,59)(37,60)(38,43)(39,54)(41,61)(42,62)(48,56)(49,63)(50,64);(1,5,7,26)(2,10,13,12)(3,16,19,18)(4,21,24,23)(6,25,29,48)(8,31,33,32)(9,34,36,20)(11,37,39,27)(14,41,42,38)(15,44,47,46)(17,28,50,49)(22,53,57,56)(30,58,59,51)(35,60,54,52)(40,61,62,43)(45,55,64,63)Polycyclic presentation:
2^9|a^2=g,d^2=i,e^2=h,f^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,e^a=eh,h^a=hi,e^b=ehi,g^b=gi,h^b=hi,f^c=fi,g^c=gi80 conjugacy classes — view the full character table.
| maximal subgroup of | #72280192, #72280557, #72280993, … |
| maximal normal subgroup of | #72280192, #72280557, #72280993, … |
| maximal quotient of | #72310123, #72321635, #72322981, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 400754);