Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #495737 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #21452 |
| commutator | C2×C8 | C23×C4 |
| Frattini, cosocle | C22×C8 | C24 |
| prime radical | #33540 | C2 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C22×C4 ◁ #2875 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
1011 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 64 points:
(1,2,8,14,9,16,7,15)(3,17,21,43,22,45,32,41)(4,23,26,11,27,39,31,40)(5,12,6,30,29,13,28,42)(10,19,35,20,36,52,34,53)(18,46,50,63,51,62,49,44)(24,55,25,37,57,56,58,38)(33,59,54,47,61,48,60,64);(1,3)(2,10)(4,18)(5,20)(6,19)(7,21)(8,32)(9,22)(11,33)(12,41)(13,43)(14,34)(15,35)(16,36)(17,42)(23,54)(24,48)(25,47)(26,49)(27,51)(28,52)(29,53)(30,45)(31,50)(37,62)(38,46)(39,60)(40,61)(44,55)(56,63)(57,59)(58,64);(1,4,9,27)(2,11,16,40)(3,18,22,51)(5,24,29,57)(6,25,28,58)(7,31,8,26)(10,33,36,61)(12,37,13,38)(14,39,15,23)(17,44,45,63)(19,47,52,64)(20,48,53,59)(21,50,32,49)(30,56,42,55)(34,60,35,54)(41,62,43,46);(1,5,8,6,9,29,7,28)(2,12,14,30,16,13,15,42)(3,19,21,20,22,52,32,53)(4,24,26,25,27,57,31,58)(10,17,35,43,36,45,34,41)(11,37,39,56,40,38,23,55)(18,47,50,48,51,64,49,59)(33,44,54,46,61,63,60,62)Polycyclic presentation:
2^9|a^2=g,c^2=i,d^2=g,e^2=h,f^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,e^a=eh,f^a=fi,h^a=hi,d^b=dh,e^b=ehi,g^b=gi,h^b=hi,f^c=fi,e^d=ei65 conjugacy classes — view the full character table.
| maximal subgroup of | #72250066, #72280389, C2×#495740, … |
| maximal normal subgroup of | #72250066, #72280389, C2×#495740, … |
| maximal quotient of | #72250645, #72280679, C2×#495740, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 402936);