Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6667772 |
| 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 | #42906 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2392 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1469 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 48 points:
(2,6)(3,4)(9,10)(11,15)(12,16)(13,14)(17,18,22,27)(19,28,31,24)(20,32,35,25)(21,36,23,26)(29,39,47,45)(30,40,37,41)(33,42,46,48)(34,43,38,44);(1,2)(3,7)(4,8)(5,6)(17,19,22,31)(18,24,27,28)(20,33,35,46)(21,30,23,37)(25,42,32,48)(26,40,36,41)(29,38,47,34)(39,43,45,44);(1,3,5,4)(2,7,6,8)(10,14)(15,16)(17,20,22,35)(18,25,27,32)(19,29,31,47)(21,34,23,38)(24,39,28,45)(26,43,36,44)(30,46,37,33)(40,42,41,48);(1,4,5,3)(2,8,6,7)(9,11,13,12)(10,15,14,16)(17,21,22,23)(18,26,27,36)(19,30,31,37)(20,34,35,38)(24,40,28,41)(25,43,32,44)(29,46,47,33)(39,42,45,48);(2,6)(7,8)(9,12,13,11)(10,16,14,15)(18,27)(19,30)(20,35)(21,23)(24,40)(28,41)(29,33)(31,37)(39,48)(42,45)(43,44)(46,47)Polycyclic presentation:
2^9|a^2=i,b^2=i,c^2=f,d^2=g,e^2=fg,h^2=i,b^a=bf,c^a=cg,d^a=df,e^a=ei,c^b=ch,e^b=efh,h^b=hi,e^c=ei,h^c=hi,e^d=ei,h^e=hi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6646616);