Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6535823 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C26×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2295 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1342 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 5 generators on 48 points:
(1,2,5,7)(3,8,4,6)(9,10,14,13)(11,16,12,15)(17,18)(19,24)(20,32)(21,26)(22,37)(23,27)(25,34)(28,45)(29,39)(30,40)(31,41)(33,48)(35,44)(36,43)(38,46)(42,47);(1,3,5,4)(2,6,7,8)(9,11,14,12)(10,15,13,16)(17,19,23,31)(18,24,27,41)(20,33,36,28)(21,29,22,30)(25,42,44,38)(26,39,37,40)(32,48,43,45)(34,47,35,46);(1,4,5,3)(2,6,7,8)(9,10,14,13)(11,16,12,15)(17,20,22,34,23,36,21,35)(18,25,26,32,27,44,37,43)(19,28,30,46,31,33,29,47)(24,38,39,45,41,42,40,48);(9,12,14,11)(10,15,13,16)(17,21,23,22)(18,26,27,37)(19,29,31,30)(20,34,36,35)(24,39,41,40)(25,43,44,32)(28,46,33,47)(38,48,42,45);(9,13,14,10)(11,16,12,15)(17,21,23,22)(18,26,27,37)(19,30,31,29)(20,35,36,34)(24,40,41,39)(25,32,44,43)(28,46,33,47)(38,48,42,45)Polycyclic presentation:
2^9|a^2=f,b^2=fi,c^2=fg,d^2=h,e^2=h,g^2=i,b^a=bf,c^a=cg,g^a=gi,c^b=ch,e^b=ei,d^c=di,e^d=ehi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6467925);