Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #4861280 |
| 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 | |
1302 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)(3,5,8,7)(11,15)(14,16)(18,22)(23,28)(24,29)(30,32)(33,34)(35,39)(36,43)(37,38)(40,45)(41,48)(42,46)(44,47);(1,2)(3,5)(4,6)(7,8)(9,10,13,12)(11,14,15,16)(19,25)(23,28)(26,31)(30,32)(34,38)(39,46)(40,43)(47,48);(9,10,13,12)(11,14,15,16)(17,18)(19,23)(20,24)(21,22)(25,28)(26,30)(27,29)(31,32)(33,34)(35,39)(36,40)(37,38)(41,47)(42,46)(43,45)(44,48);(1,2)(3,7)(4,6)(5,8)(9,11)(10,14)(12,16)(13,15)(17,19)(18,23)(20,26)(21,25)(22,28)(24,30)(27,31)(29,32)(33,35,37,42)(34,39,38,46)(36,44,45,41)(40,48,43,47);(1,3)(2,5)(4,8)(6,7)(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)(34,40,38,43)(35,41,42,44)(39,47,46,48)Polycyclic presentation:
2^9|a^2=fh,b^2=fhi,c^2=fhi,d^2=h,e^2=fgi,b^a=bf,c^a=cg,d^a=di,e^a=eh,c^b=ch,d^b=dg,e^d=ef68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 4835699);