Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6573451 |
| 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 | #36015 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2392 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1686 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)(3,7)(4,6)(5,8)(9,10)(11,14)(12,16)(13,15)(17,18,22,23)(19,27,30,24)(20,25,33,37)(21,26,36,35)(28,31,43,38)(29,41,45,39)(32,40,47,34)(42,46,48,44);(1,3,5,4)(2,6,8,7)(9,11,13,12)(10,14,15,16)(17,19,22,30)(18,24,23,27)(20,31,33,38)(21,29,36,45)(25,28,37,43)(26,39,35,41)(32,46,47,44)(34,48,40,42);(1,3,5,4)(2,7,8,6)(10,15)(14,16)(17,20,22,33)(18,25,23,37)(19,28,30,43)(21,34,36,40)(24,38,27,31)(26,32,35,47)(29,44,45,46)(39,48,41,42);(9,12,13,11)(10,16,15,14)(17,21)(18,26)(19,29)(20,32)(22,36)(23,35)(24,39)(25,40)(27,41)(28,42)(30,45)(31,46)(33,47)(34,37)(38,44)(43,48);(1,4,5,3)(2,6,8,7)(18,23)(19,27)(21,35)(24,30)(25,37)(26,36)(28,31)(32,40)(34,47)(38,43)(39,41)(42,48)Polycyclic presentation:
2^9|a^2=i,b^2=fg,c^2=f,d^2=g,e^2=fi,h^2=i,b^a=bf,c^a=cg,e^a=ei,c^b=ch,e^b=eh,h^b=hi,d^c=dh,h^c=hi,e^d=eh,h^d=hi,h^e=hi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6497623);