Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #359903 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #13869 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| prime radical | C22×2−1+4 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
2021 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)(3,14,19,9)(4,20,24,37)(5,25,28,11)(6,13)(7,29)(8,12)(10,23,36,30)(15,34)(16,32)(17,39,18,33)(21,31)(22,50)(26,54,27,38)(35,51)(40,47)(41,46)(42,62,63,59)(43,57)(44,58)(45,60)(48,52)(49,61,64,56)(53,55);(1,3)(2,9)(4,21)(5,16)(6,18)(7,17)(8,19)(10,34)(11,32)(12,14)(13,39)(15,23)(20,40)(22,49)(24,47)(25,41)(26,45)(27,44)(28,46)(29,33)(30,43)(31,37)(35,59)(36,57)(38,60)(42,51)(48,61)(50,56)(52,64)(53,62)(54,58)(55,63);(1,4)(2,10,12,36)(3,15)(5,22)(6,23)(7,30)(8,24)(9,31,14,40)(11,35,25,53)(13,20,29,37)(16,42)(17,21)(18,47)(19,43)(26,51)(27,55)(28,52)(32,56,41,61)(33,57,39,34)(38,48,54,50)(44,49)(45,64)(46,63)(58,62,60,59);(1,5,8,28)(2,11,12,25)(3,16,19,46)(4,22,24,52)(6,26,7,27)(9,32,14,41)(10,35,36,53)(13,38,29,54)(15,42,43,63)(17,44,18,45)(20,48,37,50)(21,49,47,64)(23,51,30,55)(31,56,40,61)(33,58,39,60)(34,59,57,62)Polycyclic presentation:
2^9|a^2=h,c^2=e,d^2=i,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=de,e^a=ei,g^a=gi,c^b=cg,f^b=fi,g^b=gi,h^b=hi,g^c=gi,h^c=hi,h^d=hi77 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 267100);