Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #4379698 |
| automorphism | Aut(G) order 2097152 |
| 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 | |
1402 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 5 generators on 48 points:
(2,5)(3,7)(11,15)(13,16)(17,18,21,24)(19,22,25,29)(20,27,28,23)(26,32,31,30)(34,38)(36,44)(39,41)(42,48);(19,25)(22,29)(26,31)(30,32)(33,34,37,38)(35,41,43,39)(36,40,44,45)(42,47,48,46);(1,2)(3,6)(4,5)(7,8)(9,10,12,14)(11,13,15,16)(17,19,21,25)(18,22,24,29)(20,26,28,31)(23,30,27,32)(33,34)(35,39)(36,45)(37,38)(40,44)(41,43)(42,47)(46,48);(1,3,4,7)(2,6,5,8)(9,11)(10,13)(12,15)(14,16)(17,18)(19,22)(20,23)(21,24)(25,29)(26,30)(27,28)(31,32)(33,35,37,43)(34,39,38,41)(36,42,44,48)(40,46,45,47);(17,20,21,28)(18,23,24,27)(19,26,25,31)(22,30,29,32)(33,36)(34,40)(35,42)(37,44)(38,45)(39,46)(41,47)(43,48)Polycyclic presentation:
2^9|a^2=fh,b^2=f,c^2=fgi,d^2=g,e^2=fh,b^a=bf,c^a=cg,d^a=di,e^a=eh,c^b=ch,d^b=df,e^c=ef80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 4294779);