Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #4417862 |
| automorphism | Aut(G) order 4194304 |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C24 | C25 |
| prime radical | C2×#2635 | C2 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
3658 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 40 points:
(2,4)(5,6,7,8)(11,17)(12,19)(15,20)(16,21)(25,26,30,27)(28,33,36,31)(29,34,38,32)(35,39,37,40);(10,14)(15,20)(16,21)(22,24)(26,27)(31,33)(32,34)(39,40);(5,6)(7,8)(9,10)(11,15)(12,16)(13,14)(17,20)(18,22)(19,21)(23,24)(25,27,30,26)(28,33,36,31)(29,34,38,32)(35,40,37,39);(1,2)(3,4)(5,6)(7,8)(9,11,13,17)(10,15,14,20)(12,18,19,23)(16,22,21,24)(25,28,30,36)(26,31,27,33)(29,37,38,35)(32,40,34,39);(9,12)(10,16)(11,18)(13,19)(14,21)(15,22)(17,23)(20,24)(25,29,30,38)(26,32,27,34)(28,35,36,37)(31,39,33,40)Polycyclic presentation:
2^9|a^2=fg,c^2=f,d^2=h,e^2=f,b^a=bf,c^a=cg,d^a=di,e^a=eh,c^b=ch,e^d=ef80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 4326231);