Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6667980 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C26×C4 |
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 | |
1487 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)(11,13,15,16)(17,18,22,26)(19,28,30,24)(20,33,36,43)(21,27,23,37)(25,38,42,35)(29,45,34,48)(31,44,32,40)(39,46,41,47);(1,2)(3,6)(4,8)(5,7)(9,11)(10,13)(12,15)(14,16)(17,19)(18,24)(20,34)(21,31)(22,30)(23,32)(25,41)(26,28)(27,40)(29,36)(33,45)(35,47)(37,44)(38,46)(39,42)(43,48);(11,15)(13,16)(17,20,22,36)(18,25,26,42)(19,29,30,34)(21,35,23,38)(24,39,28,41)(27,43,37,33)(31,46,32,47)(40,45,44,48);(1,3,5,4)(2,6,7,8)(18,26)(19,30)(25,42)(27,37)(29,34)(31,32)(33,43)(46,47);(1,4,5,3)(2,6,7,8)(11,15)(13,16)(17,21,22,23)(18,27,26,37)(19,31,30,32)(20,35,36,38)(24,40,28,44)(25,43,42,33)(29,46,34,47)(39,45,41,48)Polycyclic presentation:
2^9|a^2=fhi,c^2=i,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=efh,g^c=gi71 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6656438);