Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6667580 |
| 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 | C2×#553 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2395 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1552 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 48 points:
(1,2,5,7)(3,8,4,6)(9,10,12,14)(11,15,16,13)(17,18,22,23)(19,27,30,24)(20,25,33,37)(21,34,36,26)(28,31,44,38)(29,39,46,42)(32,41,35,40)(43,48,45,47);(1,2)(3,6)(4,8)(5,7)(9,11)(10,13)(12,16)(14,15)(17,19,22,30)(18,24,23,27)(20,31,33,38)(21,29,36,46)(25,28,37,44)(26,39,34,42)(32,47,35,48)(40,43,41,45);(10,14)(13,15)(17,20,22,33)(18,25,23,37)(19,28,30,44)(21,35,36,32)(24,38,27,31)(26,41,34,40)(29,45,46,43)(39,47,42,48);(1,3,5,4)(2,6,7,8)(10,14)(13,15)(17,21,22,36)(18,26,23,34)(19,29,30,46)(20,32,33,35)(24,39,27,42)(25,40,37,41)(28,43,44,45)(31,47,38,48);(1,4,5,3)(2,6,7,8)(11,16)(13,15)(18,23)(19,24)(20,33)(26,34)(27,30)(28,31)(29,39)(32,35)(38,44)(42,46)(43,47)(45,48)Polycyclic presentation:
2^9|a^2=f,b^2=i,c^2=i,d^2=fg,e^2=fgi,h^2=i,b^a=bf,c^a=cg,d^a=df,e^a=ei,c^b=ch,e^b=efh,h^b=hi,d^c=di,e^c=ei,h^c=hi,h^e=hi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6649389);