Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #493391 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | C2×#1781 |
| commutator | C24 | C23×C4 |
| Frattini, cosocle | C22×D4 | C24 |
| prime radical | #62859 | C2 |
| LCS | G ▷ C24 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C22×C4 ◁ C22×2−1+4 ◁ G | |
| LECS | G ▷ C22×D4 ▷ C22 ▷ C2 ▷ C1 | |
3255 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)(3,13)(4,9)(5,10)(6,24,26,11)(7,12)(8,16,29,48)(14,40)(15,38)(17,39)(18,30)(19,36,25,31)(20,32)(21,33)(22,51,52,34)(23,35)(27,42,54,47)(28,45,56,64)(37,43)(41,60)(44,59)(46,61)(49,58,50,57)(53,62,55,63);(1,3)(2,8)(4,14)(5,15)(6,16)(7,17)(9,27)(10,28)(11,13)(12,29)(18,41)(19,42)(20,43)(21,44)(22,45)(23,46)(24,39)(25,47)(26,48)(30,53)(31,40)(32,54)(33,55)(34,38)(35,56)(36,37)(49,62)(50,63)(51,61)(52,64)(57,60)(58,59);(1,4,7,20)(2,9,12,32)(3,14,17,43)(5,21,23,18)(6,25,26,19)(8,27,29,54)(10,33,35,30)(11,36,24,31)(13,37,39,40)(15,44,46,41)(16,47,48,42)(22,49,52,50)(28,55,56,53)(34,57,51,58)(38,60,61,59)(45,62,64,63);(1,5,7,23)(2,10,12,35)(3,15,17,46)(4,18,20,21)(6,22,26,52)(8,28,29,56)(9,30,32,33)(11,34,24,51)(13,38,39,61)(14,41,43,44)(16,45,48,64)(19,49,25,50)(27,53,54,55)(31,57,36,58)(37,59,40,60)(42,62,47,63)Polycyclic presentation:
2^9|a^2=g,c^2=i,d^2=i,b^a=be,c^a=cf,e^a=eh,h^a=hi,f^b=fi,g^b=gh,d^c=di,e^c=ei,g^e=gi65 conjugacy classes — view the full character table.
| maximal subgroup of | #72302902, #72302905, #72302908, … |
| maximal normal subgroup of | #72302902, #72302905, #72302908, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 400589);