Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #6131635 |
| automorphism | Aut(G) order 6291456 |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C24×C42 |
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 | |
1638 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 48 points:
(2,5)(3,7)(9,10,12,14)(11,15,16,13)(18,22)(19,25)(20,27)(30,32)(35,41)(39,44)(42,47)(46,48);(2,5)(6,8)(9,11,12,16)(10,13,14,15)(20,27)(24,29)(26,31)(30,32)(34,38)(35,41)(40,45)(42,47);(17,18)(19,23)(20,24)(21,22)(25,28)(26,30)(27,29)(31,32)(33,34)(35,39)(36,40)(37,38)(41,44)(42,46)(43,45)(47,48);(1,2)(3,6)(4,5)(7,8)(9,11)(10,13)(12,16)(14,15)(17,19,21,25)(18,23,22,28)(20,26,27,31)(24,30,29,32)(33,35,37,41)(34,39,38,44)(36,42,43,47)(40,46,45,48);(1,3)(2,6)(4,7)(5,8)(9,10)(11,13)(12,14)(15,16)(17,20)(18,24)(19,26)(21,27)(22,29)(23,30)(25,31)(28,32)(33,36,37,43)(34,40,38,45)(35,42,41,47)(39,46,44,48)Polycyclic presentation:
2^9|a^2=f,b^2=f,d^2=gh,e^2=h,b^a=bf,c^a=cg,d^a=di,e^a=ehi,c^b=ch,d^b=dgi,e^b=eg68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6039721);