Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #33607 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C23×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C22×D8 |
| commutator | C2×C8 | C23×C4 |
| Frattini, cosocle | C22×C8 | C24 |
| socle | C22 | C2×C4×D8 |
| prime radical | #3147 | C22 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C2×#97 ◁ #1726 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
558 conjugacy classes of subgroups.
| minimal degree | μ(G) = 72 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 72 points:
(1,2)(3,6)(4,5)(7,8)(9,10,15,22,17,23,16,40)(11,24,29,44,31,54,30,43)(12,32,35,62,37,63,36,47)(13,20,38,50,39,51,14,21)(18,41,45,26,46,28,53,27)(19,33,48,64,49,65,66,34)(25,42,58,69,60,70,59,71)(52,56,61,67,72,55,68,57);(2,5)(6,8)(9,11,14,28,16,30,39,26,17,31,38,41,15,29,13,27)(10,18,20,44,22,45,50,54,23,46,51,43,40,53,21,24)(12,25,34,57,36,59,65,55,37,60,64,67,35,58,33,56)(19,42,32,61,48,69,62,72,49,70,63,68,66,71,47,52);(1,3,4,7)(2,6,5,8)(9,12,13,33,15,35,38,64,17,37,39,65,16,36,14,34)(10,19,21,47,40,66,51,63,23,49,50,62,22,48,20,32)(11,25,27,56,29,58,41,67,31,60,26,55,30,59,28,57)(18,42,24,52,53,71,43,68,46,70,54,72,45,69,44,61);(9,13,15,38,17,39,16,14)(10,20,22,50,23,51,40,21)(11,26,29,28,31,27,30,41)(12,33,35,64,37,65,36,34)(18,43,45,24,46,44,53,54)(19,32,48,62,49,63,66,47)(25,55,58,57,60,56,59,67)(42,68,69,52,70,61,71,72)Polycyclic presentation:
2^9|a^2=g,b^2=f,c^2=e,d^2=g,e^2=hi,f^2=h,g^2=i,h^2=i,b^a=be,c^a=cf,e^a=eh,f^a=fhi,h^a=hi,d^b=di104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 403253);