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)(9,10,15,22,17,23,16,41)(11,24,29,45,31,56,30,44)(12,32,36,64,38,65,37,49)(13,20,39,52,40,53,14,21)(18,42,46,26,47,28,55,27)(19,34,50,57,51,67,68,35)(25,43,61,70,33,48,62,72)(54,59,69,66,71,58,63,60);(2,5)(6,8)(9,11,14,28,16,30,40,26,17,31,39,42,15,29,13,27)(10,18,20,45,22,46,52,56,23,47,53,44,41,55,21,24)(12,33,35,66,37,61,67,59,38,25,57,60,36,62,34,58)(19,48,32,63,50,72,64,54,51,43,65,69,68,70,49,71);(1,3,4,7)(2,6,5,8)(9,12,13,34,15,36,39,57,17,38,40,67,16,37,14,35)(10,19,21,49,41,68,53,65,23,51,52,64,22,50,20,32)(11,25,27,59,29,61,42,66,31,33,26,58,30,62,28,60)(18,43,24,54,55,72,44,63,47,48,56,71,46,70,45,69);(9,13,15,39,17,40,16,14)(10,20,22,52,23,53,41,21)(11,26,29,28,31,27,30,42)(12,34,36,57,38,67,37,35)(18,44,46,24,47,45,55,56)(19,32,50,64,51,65,68,49)(25,58,61,60,33,59,62,66)(43,63,70,54,48,69,72,71)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,c^b=ci,d^b=di104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 403254);