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 | |
1377 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 48 points:
(1,2,7,6)(3,5,8,4)(11,15)(13,16)(18,22)(23,28)(24,29)(30,32)(33,34)(35,38)(36,39)(37,42)(40,46)(41,43)(44,48)(45,47);(1,3)(2,4)(5,6)(7,8)(9,10,12,14)(11,13,15,16)(19,25)(23,28)(26,31)(30,32)(34,35)(36,43)(40,42)(44,45);(1,4,7,5)(2,8,6,3)(17,18)(19,23)(20,24)(21,22)(25,28)(26,30)(27,29)(31,32)(33,35,38,34)(36,41,43,39)(37,42,46,40)(44,48,45,47);(1,5,7,4)(2,8,6,3)(9,11)(10,13)(12,15)(14,16)(17,19,21,25)(18,23,22,28)(20,26,27,31)(24,30,29,32)(33,36,38,43)(34,39,35,41)(37,45,46,44)(40,48,42,47);(1,6,7,2)(3,5,8,4)(17,20,21,27)(18,24,22,29)(19,26,25,31)(23,30,28,32)(33,37)(34,40)(35,42)(36,44)(38,46)(39,47)(41,48)(43,45)Polycyclic presentation:
2^9|a^2=fh,b^2=i,c^2=f,d^2=g,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, 4781353);