Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #4861280 |
| 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 | |
1441 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 48 points:
(2,3)(4,8)(10,14)(12,15)(18,22)(19,25)(24,29)(26,31)(33,34,38,40)(35,41,43,36)(37,45,47,39)(42,44,48,46);(2,3)(6,7)(10,14)(11,16)(19,25)(23,28)(26,31)(30,32)(33,35,38,43)(34,36,40,41)(37,42,47,48)(39,44,45,46);(1,2,5,3)(4,6,8,7)(9,10)(11,15)(12,16)(13,14)(17,18,21,22)(19,23,25,28)(20,24,27,29)(26,30,31,32)(33,35)(34,36)(37,42)(38,43)(39,44)(40,41)(45,46)(47,48);(1,3,5,2)(4,7,8,6)(9,11,13,16)(10,15,14,12)(17,19)(18,23)(20,26)(21,25)(22,28)(24,30)(27,31)(29,32)(33,36)(34,35)(37,46)(38,41)(39,48)(40,43)(42,45)(44,47);(1,4,5,8)(2,6,3,7)(9,12,13,15)(10,16,14,11)(17,20,21,27)(18,24,22,29)(19,26,25,31)(23,30,28,32)(33,37,38,47)(34,39,40,45)(35,42,43,48)(36,44,41,46)Polycyclic presentation:
2^9|a^2=f,b^2=f,c^2=i,d^2=fh,e^2=g,b^a=bf,c^a=cg,d^a=di,e^a=eh,c^b=ch,d^b=dg,e^d=ef68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 4787119);