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 | |
1316 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 5 generators on 48 points:
(1,2,6,4)(3,7,5,8)(9,10,13,11)(12,15,16,14)(17,18,21,24)(19,22,27,25)(20,23,28,30)(26,29,32,31)(33,34)(35,39)(36,38)(37,44)(40,46)(41,43)(42,48)(45,47);(1,3,6,5)(2,7,4,8)(9,11,13,10)(12,15,16,14)(20,28)(23,30)(26,32)(29,31)(34,36)(35,41)(40,44)(42,45);(1,4,6,2)(3,7,5,8)(17,18)(19,25)(20,23)(21,24)(22,27)(26,31)(28,30)(29,32)(33,35,38,41)(34,39,36,43)(37,45,46,42)(40,48,44,47);(9,12)(10,14)(11,15)(13,16)(17,19)(18,22)(20,26)(21,27)(23,29)(24,25)(28,32)(30,31)(33,36,38,34)(35,39,41,43)(37,44,46,40)(42,47,45,48);(1,5,6,3)(2,7,4,8)(17,20)(18,23)(19,26)(21,28)(22,29)(24,30)(25,31)(27,32)(33,37,38,46)(34,40,36,44)(35,42,41,45)(39,47,43,48)Polycyclic presentation:
2^9|a^2=ghi,b^2=hi,c^2=h,d^2=f,e^2=h,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, 4986163);