Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6574024 |
| 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 | #2414 | 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:
(1,2,5,7)(3,8,4,6)(9,10)(11,14)(12,15)(13,16)(17,18,23,27)(19,28,32,44)(20,25,34,41)(21,26,22,37)(24,30,39,31)(29,43,33,47)(35,42,36,48)(38,45,40,46);(9,11)(10,13)(12,16)(14,15)(17,19,21,30,23,32,22,31)(18,24,37,44,27,39,26,28)(20,33,35,46,34,29,36,45)(25,40,48,43,41,38,42,47);(1,3,5,4)(2,6,7,8)(11,16)(13,14)(17,20,23,34)(18,25,27,41)(19,29,32,33)(21,35,22,36)(24,38,39,40)(26,42,37,48)(28,43,44,47)(30,45,31,46);(1,4,5,3)(2,6,7,8)(9,11)(10,14)(12,16)(13,15)(20,34)(25,41)(29,33)(35,36)(38,40)(42,48)(43,47)(45,46);(11,16)(13,14)(17,21,23,22)(18,26,27,37)(19,30,32,31)(20,35,34,36)(24,28,39,44)(25,42,41,48)(29,45,33,46)(38,43,40,47)Polycyclic presentation:
2^9|a^2=gi,b^2=fh,c^2=gi,d^2=g,e^2=i,f^2=i,b^a=bf,c^a=cg,f^a=fi,c^b=ch,e^b=ehi,d^c=dh,e^d=ehi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6521903);