Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #3069037 |
| 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 | |
1451 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)(17,18,21,24)(19,22,25,29)(20,27,28,23)(26,32,31,30)(33,34)(35,36)(37,41)(38,39)(40,43)(42,44)(45,47)(46,48);(9,10,13,12)(11,15,16,14)(19,25)(22,29)(26,31)(30,32)(34,39)(36,40)(37,45)(42,48);(1,2,5,4)(3,6,7,8)(9,11,13,16)(10,14,12,15)(17,18)(19,22)(20,23)(21,24)(25,29)(26,30)(27,28)(31,32)(33,35,38,43)(34,40,39,36)(37,42,45,48)(41,46,47,44);(1,3,5,7)(2,6,4,8)(17,19)(18,22)(20,26)(21,25)(23,30)(24,29)(27,32)(28,31)(33,36,38,40)(34,35,39,43)(37,46,45,44)(41,48,47,42);(1,4,5,2)(3,8,7,6)(9,12,13,10)(11,15,16,14)(17,20,21,28)(18,23,24,27)(19,26,25,31)(22,30,29,32)(33,37,38,45)(34,41,39,47)(35,42,43,48)(36,44,40,46)Polycyclic presentation:
2^9|a^2=fg,b^2=h,c^2=fhi,d^2=fi,e^2=ghi,b^a=bf,c^a=cg,d^a=di,e^a=efg,c^b=ch,d^b=dg,e^b=ef,d^c=df,e^d=ef74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 3230230);