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 | |
1463 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 48 points:
(1,2,5,7)(3,8,4,6)(9,10,13,15)(11,16,14,12)(18,22)(23,28)(24,29)(30,32)(33,34)(35,36)(37,41)(38,39)(40,43)(42,44)(45,48)(46,47);(1,3)(2,6)(4,5)(7,8)(9,11,13,14)(10,12,15,16)(19,25)(23,28)(26,31)(30,32)(34,39)(36,40)(41,47)(44,45);(1,2)(3,8)(4,6)(5,7)(9,10)(11,16)(12,14)(13,15)(17,18,21,22)(19,23,25,28)(20,24,27,29)(26,30,31,32)(33,35)(34,40)(36,39)(37,42)(38,43)(41,45)(44,47)(46,48);(1,4,5,3)(2,6,7,8)(9,12,13,16)(10,14,15,11)(17,19)(18,23)(20,26)(21,25)(22,28)(24,30)(27,31)(29,32)(33,36,38,40)(34,35,39,43)(37,45,46,44)(41,48,47,42);(1,2)(3,6)(4,8)(5,7)(9,10)(11,12)(13,15)(14,16)(17,20,21,27)(18,24,22,29)(19,26,25,31)(23,30,28,32)(33,37,38,46)(34,41,39,47)(35,42,43,48)(36,44,40,45)Polycyclic presentation:
2^9|a^2=h,b^2=i,c^2=fg,d^2=f,e^2=gh,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, 4781296);