Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #5473759 |
| automorphism | Aut(G) order 2097152 |
| 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 | |
1602 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)(3,5)(4,6)(7,8)(10,12)(14,15)(19,23)(22,24)(25,26,30,34)(27,32,28,31)(29,33,38,37)(35,40,36,39);(4,8)(6,7)(10,12)(14,15)(18,21)(22,24)(25,27,30,28)(26,31,34,32)(29,35,38,36)(33,39,37,40);(2,3)(6,7)(9,10,13,12)(11,15,16,14)(17,18)(19,22)(20,21)(23,24)(25,26)(27,32)(28,31)(29,37)(30,34)(33,38)(35,39)(36,40);(1,3,5,2)(4,7,8,6)(9,11,13,16)(10,14,12,15)(25,28,30,27)(26,32,34,31)(29,36,38,35)(33,40,37,39);(1,4,5,8)(2,6,3,7)(9,12,13,10)(11,15,16,14)(17,19)(18,22)(20,23)(21,24)(25,29)(26,33)(27,35)(28,36)(30,38)(31,39)(32,40)(34,37)Polycyclic presentation:
2^9|a^2=f,b^2=f,c^2=gi,d^2=g,e^2=fg,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, 5398213);