Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #5035983 |
| 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 | |
1489 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 48 points:
(1,2,5,6)(3,4,7,8)(9,10)(11,15)(12,14)(13,16)(17,18,21,22)(19,23,25,29)(20,27,28,24)(26,32,31,30)(33,34)(35,36)(37,41)(38,39)(40,43)(42,44)(45,47)(46,48);(1,3,5,7)(2,4,6,8)(9,10)(11,15)(12,14)(13,16)(18,22)(19,25)(20,28)(30,32)(34,39)(36,40)(37,45)(42,46);(2,6)(3,7)(9,10,12,14)(11,13,16,15)(17,18)(19,23)(20,27)(21,22)(24,28)(25,29)(26,32)(30,31)(33,35)(34,40)(36,39)(37,46)(38,43)(41,44)(42,45)(47,48);(1,2)(3,4)(5,6)(7,8)(9,10)(11,13)(12,14)(15,16)(17,19,21,25)(18,23,22,29)(20,26,28,31)(24,30,27,32)(33,36,38,40)(34,35,39,43)(37,44,45,48)(41,42,47,46);(1,4)(2,3)(5,8)(6,7)(9,11,12,16)(10,13,14,15)(17,20)(18,24)(19,26)(21,28)(22,27)(23,30)(25,31)(29,32)(33,37,38,45)(34,41,39,47)(35,42,43,46)(36,44,40,48)Polycyclic presentation:
2^9|a^2=fghi,b^2=i,c^2=fgi,d^2=f,e^2=fhi,b^a=bf,c^a=cg,d^a=di,e^a=eh,c^b=ch,d^b=df,e^b=eg,d^c=dfg,e^c=ef68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 5113574);