Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6907909 |
| 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 | #2407 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2290 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1495 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)(3,6)(4,5)(7,8)(17,18,22,25)(19,28,30,23)(20,32,33,40)(21,26,35,27)(24,39,41,34)(29,43,45,47)(31,38,46,37)(36,44,42,48);(1,3,7,4)(2,5,8,6)(9,10)(11,15)(12,14)(13,16)(17,19,22,30)(18,23,25,28)(20,29,33,45)(21,31,35,46)(24,36,41,42)(26,37,27,38)(32,47,40,43)(34,44,39,48);(1,4,7,3)(2,5,8,6)(9,11)(10,13)(12,16)(14,15)(17,20,21,34,22,33,35,39)(18,24,27,32,25,41,26,40)(19,29,31,44,30,45,46,48)(23,36,38,47,28,42,37,43);(1,5,7,6)(2,3,8,4)(11,16)(13,15)(18,25)(19,30)(20,33)(26,27)(31,46)(34,39)(36,42)(43,47);(1,6,7,5)(2,3,8,4)(10,14)(13,15)(17,21,22,35)(18,26,25,27)(19,31,30,46)(20,34,33,39)(23,37,28,38)(24,40,41,32)(29,44,45,48)(36,43,42,47)Polycyclic presentation:
2^9|a^2=i,b^2=f,c^2=fgi,d^2=fi,e^2=f,g^2=i,b^a=bf,c^a=cg,d^a=df,g^a=gi,c^b=ch,d^b=di,e^b=efhi,d^c=dfh68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6822883);