Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #5956507 |
| 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 | |
1433 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)(3,4)(5,6)(7,8)(11,15)(13,16)(17,18,22,19)(20,25,27,23)(21,26,29,24)(28,30,32,31)(33,34,38,39)(35,37,43,40)(36,44,46,41)(42,47,48,45);(1,3)(2,4)(5,7)(6,8)(9,10,12,14)(11,13,15,16)(20,27)(23,25)(28,32)(30,31)(34,39)(37,40)(41,44)(45,47);(2,6)(3,7)(9,10)(11,13)(12,14)(15,16)(17,19,22,18)(20,25,27,23)(21,26,29,24)(28,31,32,30)(33,35)(34,40)(36,42)(37,39)(38,43)(41,47)(44,45)(46,48);(1,2)(3,8)(4,7)(5,6)(17,20)(18,23)(19,25)(21,28)(22,27)(24,30)(26,31)(29,32)(33,36,38,46)(34,41,39,44)(35,42,43,48)(37,45,40,47);(1,4,5,8)(2,7,6,3)(9,11,12,15)(10,13,14,16)(17,21)(18,24)(19,26)(20,28)(22,29)(23,30)(25,31)(27,32)(33,37,38,40)(34,35,39,43)(36,45,46,47)(41,42,44,48)Polycyclic presentation:
2^9|a^2=i,b^2=fgh,c^2=fi,d^2=f,e^2=fh,b^a=bf,c^a=cg,d^a=di,e^a=ehi,c^b=ch,d^b=dgi,e^b=ef,d^c=dfg62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 5942152);