Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #2760365 |
| automorphism | Aut(G) order 1048576 |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C25×C4 |
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 | |
1421 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 48 points:
(3,7)(6,8)(9,10,13,12)(11,14,15,16)(18,22)(20,27)(23,28)(26,31)(33,34,37,38)(35,39,43,41)(36,40,45,46)(42,47,44,48);(1,2,5,4)(3,6,7,8)(11,15)(14,16)(19,25)(23,28)(26,31)(30,32)(34,38)(39,41)(40,46)(47,48);(9,11,13,15)(10,14,12,16)(17,18)(19,23)(20,24)(21,22)(25,28)(26,30)(27,29)(31,32)(33,34)(35,41)(36,40)(37,38)(39,43)(42,48)(44,47)(45,46);(1,3)(2,6)(4,8)(5,7)(9,10,13,12)(11,14,15,16)(17,19,21,25)(18,23,22,28)(20,26,27,31)(24,30,29,32)(33,35,37,43)(34,39,38,41)(36,44,45,42)(40,48,46,47);(1,4,5,2)(3,8,7,6)(9,12,13,10)(11,16,15,14)(17,20,21,27)(18,24,22,29)(19,26,25,31)(23,30,28,32)(33,36,37,45)(34,40,38,46)(35,42,43,44)(39,47,41,48)Polycyclic presentation:
2^9|a^2=h,b^2=i,c^2=fh,d^2=fgh,e^2=fghi,b^a=bf,c^a=cg,d^a=di,e^a=efg,c^b=ch,d^b=dfg,d^c=df,e^d=ef74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 2711653);