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, cosocle | C24 | C25 |
| prime radical | C25 | C24 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
1408 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 48 points:
(1,2,6,5)(3,8,7,4)(11,15)(13,16)(18,22)(23,28)(24,29)(30,32)(33,34)(35,36)(37,45)(38,39)(40,44)(41,47)(42,46)(43,48);(1,3,6,7)(2,4,5,8)(9,10,12,14)(11,13,15,16)(19,25)(23,28)(26,31)(30,32)(34,39)(35,40)(41,45)(42,48);(1,4)(2,7)(3,5)(6,8)(17,18,21,22)(19,23,25,28)(20,24,27,29)(26,30,31,32)(33,35,38,40)(34,36,39,44)(37,42,47,48)(41,43,45,46);(1,4)(2,3)(5,7)(6,8)(9,11)(10,13)(12,15)(14,16)(17,19)(18,23)(20,26)(21,25)(22,28)(24,30)(27,31)(29,32)(33,36)(34,40)(35,39)(37,46)(38,44)(41,42)(43,47)(45,48);(1,5,6,2)(3,8,7,4)(17,20,21,27)(18,24,22,29)(19,26,25,31)(23,30,28,32)(33,37,38,47)(34,41,39,45)(35,42,40,48)(36,43,44,46)Polycyclic presentation:
2^9|a^2=fh,b^2=fi,c^2=fg,e^2=gh,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, 4781291);