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, prime radical, cosocle | C24 | C25 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
1306 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 5 generators on 48 points:
(1,2,4,6)(3,5,7,8)(9,10)(11,14)(12,13)(15,16)(17,18,22,19)(20,25,27,23)(21,26,29,24)(28,30,32,31)(35,39)(36,43)(42,47)(44,48);(9,11)(10,14)(12,15)(13,16)(18,19)(20,27)(24,26)(28,32)(33,34,38,41)(35,36,39,43)(37,45,46,40)(42,48,47,44);(9,11,13,16)(10,15,12,14)(17,19,22,18)(20,23,27,25)(21,24,29,26)(28,31,32,30)(33,35)(34,36)(37,42)(38,39)(40,44)(41,43)(45,48)(46,47);(1,2)(3,5)(4,6)(7,8)(9,11)(10,15)(12,14)(13,16)(17,20,22,27)(18,23,19,25)(21,28,29,32)(24,30,26,31)(33,36)(34,39)(35,41)(37,44)(38,43)(40,47)(42,45)(46,48);(1,3,4,7)(2,5,6,8)(9,12,13,10)(11,14,16,15)(17,21,22,29)(18,24,19,26)(20,28,27,32)(23,30,25,31)(33,37,38,46)(34,40,41,45)(35,42,39,47)(36,44,43,48)Polycyclic presentation:
2^9|a^2=gi,b^2=fgh,c^2=h,d^2=f,e^2=i,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, 5070133);