Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #4597856 |
| 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 | |
1436 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 5 generators on 40 points:
(1,2,5,4)(3,7,8,6)(10,14)(11,15)(18,22)(20,24)(25,26)(27,30)(28,31)(29,32)(33,35)(34,37)(36,38)(39,40);(9,10,13,14)(11,12,15,16)(18,22)(19,23)(26,27)(28,35)(32,34)(36,40);(10,14)(12,16)(17,18)(19,20)(21,22)(23,24)(25,27,30,26)(28,33,35,31)(29,32,37,34)(36,38,40,39);(1,3,5,8)(2,6,4,7)(9,11,13,15)(10,12,14,16)(17,19)(18,20)(21,23)(22,24)(25,28,30,35)(26,31,27,33)(29,36,37,40)(32,38,34,39);(1,4,5,2)(3,7,8,6)(9,12)(10,15)(11,14)(13,16)(17,20)(18,19)(21,24)(22,23)(25,29)(26,32)(27,34)(28,36)(30,37)(31,38)(33,39)(35,40)Polycyclic presentation:
2^9|a^2=ghi,b^2=gh,c^2=fgh,d^2=fghi,e^2=ghi,b^a=bf,c^a=cg,d^a=di,e^a=efh,c^b=ch,d^b=dg,e^c=ef68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 4768456);