Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #3069037 |
| 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 | |
1592 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)(6,8)(9,10,13,16)(11,15,12,14)(17,18,21,22)(19,23,24,20)(25,26)(27,28)(29,37)(30,31)(32,38)(33,36)(34,39)(35,40);(3,7)(6,8)(9,10,13,16)(11,14,12,15)(19,24)(20,23)(26,31)(27,33)(29,38)(35,39);(1,2,5,4)(3,6,7,8)(9,11,13,12)(10,14,16,15)(17,19)(18,20)(21,24)(22,23)(25,27,30,33)(26,28,31,36)(29,34,38,40)(32,35,37,39);(1,3,5,7)(2,6,4,8)(9,10)(11,14)(12,15)(13,16)(17,20)(18,19)(21,23)(22,24)(25,28)(26,27)(29,39)(30,36)(31,33)(32,40)(34,37)(35,38);(1,4,5,2)(3,8,7,6)(9,12,13,11)(10,15,16,14)(17,18)(19,20)(21,22)(23,24)(25,29)(26,32)(27,34)(28,35)(30,38)(31,37)(33,40)(36,39)Polycyclic presentation:
2^9|a^2=fhi,b^2=i,c^2=gh,d^2=fghi,e^2=fgh,b^a=bf,c^a=cg,d^a=di,e^a=efg,c^b=ch,d^b=dg,e^b=ef,d^c=df,e^d=ef74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 3075522);