Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6996639 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | #30414 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2398 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1610 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 48 points:
(1,2)(3,8)(4,6)(5,7)(9,10)(11,13)(12,14)(15,16)(17,18,21,27,22,26,35,41)(19,28,31,40,30,24,38,23)(20,32,37,46,36,25,43,45)(29,42,33,39,34,44,47,48);(9,10)(11,15)(12,14)(13,16)(17,19,22,30)(18,23,26,40)(20,33,36,47)(21,38,35,31)(24,41,28,27)(25,44,32,42)(29,43,34,37)(39,46,48,45);(1,3,5,4)(2,6,7,8)(9,11,12,16)(10,13,14,15)(17,19)(18,24)(20,34)(21,31)(22,30)(23,27)(25,44)(26,28)(29,36)(32,42)(33,43)(35,38)(37,47)(39,46)(40,41)(45,48);(11,16)(13,15)(17,20,35,43,22,36,21,37)(18,25,41,46,26,32,27,45)(19,29,38,47,30,34,31,33)(23,39,24,42,40,48,28,44);(1,4,5,3)(2,6,7,8)(10,14)(13,15)(18,26)(19,30)(20,36)(27,41)(31,38)(37,43)(39,48)(42,44)Polycyclic presentation:
2^9|a^2=f,b^2=i,c^2=gh,d^2=fi,e^2=gi,f^2=i,b^a=bf,c^a=cg,d^a=di,e^a=eh,c^b=ch,d^b=df,e^b=eh,f^b=fi,d^c=dh,e^c=ei,e^d=ei68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6998214);