Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #3332717 |
| 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 | |
1466 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)(17,18,21,24)(19,22,25,29)(20,27,28,23)(26,32,31,30)(33,34)(35,38)(36,39)(37,40)(41,43)(42,47)(44,46)(45,48);(10,14)(15,16)(19,25)(22,29)(26,31)(30,32)(33,34,37,40)(35,41,43,38)(36,39,46,44)(42,48,45,47);(1,2,5,4)(3,6,7,8)(9,10,13,14)(11,15,12,16)(17,18)(19,22)(20,23)(21,24)(25,29)(26,30)(27,28)(31,32)(33,34)(35,38)(36,44)(37,40)(39,46)(41,43)(42,48)(45,47);(1,3,5,7)(2,6,4,8)(9,11,13,12)(10,15,14,16)(17,19,21,25)(18,22,24,29)(20,26,28,31)(23,30,27,32)(33,35,37,43)(34,38,40,41)(36,45,46,42)(39,48,44,47);(1,4,5,2)(3,8,7,6)(9,12,13,11)(10,16,14,15)(17,20,21,28)(18,23,24,27)(19,26,25,31)(22,30,29,32)(33,36)(34,39)(35,42)(37,46)(38,47)(40,44)(41,48)(43,45)Polycyclic presentation:
2^9|a^2=g,b^2=f,c^2=fhi,d^2=ghi,e^2=fghi,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,d^b=dfg,e^c=ef,e^d=ef74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 3729183);