Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #5473759 |
| 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 | |
1360 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 5 generators on 40 points:
(1,2,5,7)(3,8,6,4)(9,10,14,12)(11,16,13,15)(19,23)(21,24)(26,30)(27,33)(28,36)(38,40);(1,3)(2,4)(5,6)(7,8)(9,10,14,12)(11,15,13,16)(17,18,20,22)(19,21,23,24)(26,30)(28,36)(31,34)(35,39);(1,4,5,8)(2,6,7,3)(9,11,14,13)(10,15,12,16)(25,26,29,30)(27,34,33,31)(28,32,36,37)(35,40,39,38);(1,4)(2,3)(5,8)(6,7)(9,12,14,10)(11,15,13,16)(17,19)(18,21)(20,23)(22,24)(25,27,29,33)(26,31,30,34)(28,35,36,39)(32,38,37,40);(1,2)(3,4)(5,7)(6,8)(9,13,14,11)(10,15,12,16)(25,28)(26,32)(27,35)(29,36)(30,37)(31,38)(33,39)(34,40)Polycyclic presentation:
2^9|a^2=gh,b^2=fgi,c^2=g,d^2=fg,e^2=fgh,b^a=bf,c^a=cg,d^a=di,e^a=efh,c^b=ch,e^b=efg,d^c=dg,e^c=ef68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 5393465);