Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #5024635 |
| automorphism | Aut(G) order 1048576 |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C24 | C25 |
| prime radical | C25 | C24 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
1458 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 48 points:
(3,7)(6,8)(9,10)(11,13)(12,14)(15,16)(20,27)(24,29)(25,31)(30,32)(33,34,38,39)(35,36,41,44)(37,45,48,40)(42,47,46,43);(1,2,5,4)(3,6,7,8)(18,22)(23,28)(24,29)(30,32)(34,39)(35,41)(40,45)(42,46);(9,10,12,14)(11,13,15,16)(17,18)(19,23)(20,24)(21,22)(25,30)(26,28)(27,29)(31,32)(33,35)(34,36)(37,46)(38,41)(39,44)(40,47)(42,48)(43,45);(1,3)(2,6)(4,8)(5,7)(9,10)(11,13)(12,14)(15,16)(17,19,21,26)(18,23,22,28)(20,25,27,31)(24,30,29,32)(33,36)(34,35)(37,47)(38,44)(39,41)(40,46)(42,45)(43,48);(1,4,5,2)(3,8,7,6)(9,11,12,15)(10,13,14,16)(17,20,21,27)(18,24,22,29)(19,25,26,31)(23,30,28,32)(33,37)(34,40)(35,42)(36,43)(38,48)(39,45)(41,46)(44,47)Polycyclic presentation:
2^9|a^2=f,b^2=fi,c^2=g,d^2=fh,e^2=ghi,b^a=bf,c^a=cg,d^a=di,e^a=eh,c^b=ch,d^c=dg,e^c=ef,e^d=ef80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 4943178);