Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6573591 |
| 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 | |
1462 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)(4,8)(5,6)(17,18)(19,28)(20,33)(21,26)(22,27)(23,37)(24,30)(25,36)(29,45)(31,44)(32,40)(34,48)(35,43)(38,42)(39,47)(41,46);(2,6)(7,8)(9,10,12,13)(11,14,15,16)(17,19,22,30)(18,24,27,28)(20,34,35,29)(21,31,23,32)(25,41,42,39)(26,40,37,44)(33,45,43,48)(36,47,38,46);(1,3,5,4)(2,7,6,8)(10,13)(11,15)(17,20,22,35)(18,25,27,42)(19,29,30,34)(21,36,23,38)(24,39,28,41)(26,43,37,33)(31,46,32,47)(40,45,44,48);(1,4,5,3)(2,8,6,7)(9,11)(10,14)(12,15)(13,16)(19,30)(20,35)(24,28)(25,42)(31,32)(33,43)(36,38)(40,44);(10,13)(11,15)(17,21,22,23)(18,26,27,37)(19,31,30,32)(20,36,35,38)(24,40,28,44)(25,43,42,33)(29,46,34,47)(39,45,41,48)Polycyclic presentation:
2^9|b^2=h,c^2=f,d^2=fi,e^2=i,g^2=i,b^a=bf,c^a=cg,g^a=gi,c^b=ch,d^b=di,e^b=ehi,d^c=dh,g^c=gi,e^d=ehi83 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6508667);