Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6533195 |
| 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 | |
1445 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 48 points:
(1,2,5,8)(3,7,4,6)(17,18,22,28,23,26,21,27)(19,24,32,39,30,40,31,38)(20,33,34,41,35,42,36,25)(29,43,45,47,44,48,46,37);(1,3,5,4)(2,6,8,7)(9,10)(11,14)(12,13)(15,16)(17,19,23,30)(18,24,26,40)(20,29,35,44)(21,31,22,32)(25,37,41,47)(27,38,28,39)(33,43,42,48)(34,45,36,46);(1,3,5,4)(2,7,8,6)(10,13)(14,16)(17,20,23,35)(18,25,26,41)(19,29,30,44)(21,34,22,36)(24,37,40,47)(27,33,28,42)(31,45,32,46)(38,43,39,48);(1,4,5,3)(2,6,8,7)(9,11)(10,14)(12,15)(13,16)(18,26)(19,30)(25,41)(27,28)(29,44)(31,32)(33,42)(45,46);(11,15)(14,16)(17,21,23,22)(18,27,26,28)(19,31,30,32)(20,34,35,36)(24,38,40,39)(25,33,41,42)(29,45,44,46)(37,43,47,48)Polycyclic presentation:
2^9|a^2=fg,b^2=fi,c^2=fi,d^2=f,e^2=i,g^2=i,b^a=bf,c^a=cg,d^a=di,c^b=ch,d^b=di,g^c=gi,e^d=eh77 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6460474);