Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #2760365 |
| 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 | |
1726 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 40 points:
(1,2)(3,6)(4,8)(5,7)(10,12)(11,15)(19,20)(22,23)(25,26,30,28)(27,31,33,35)(29,36,37,32)(34,40,39,38);(1,3,5,8)(2,6,7,4)(11,15)(14,16)(17,18,21,24)(19,22,20,23)(26,28)(27,33)(32,36)(34,39);(1,3)(2,4)(5,8)(6,7)(9,10,13,12)(11,14,15,16)(17,19,21,20)(18,22,24,23)(25,27)(26,31)(28,35)(29,34)(30,33)(32,38)(36,40)(37,39);(1,3)(2,4)(5,8)(6,7)(9,11)(10,14)(12,16)(13,15)(17,18)(19,22)(20,23)(21,24)(25,28,30,26)(27,31,33,35)(29,32,37,36)(34,40,39,38);(1,4)(2,3)(5,6)(7,8)(9,12,13,10)(11,16,15,14)(17,20,21,19)(18,23,24,22)(25,29,30,37)(26,32,28,36)(27,34,33,39)(31,38,35,40)Polycyclic presentation:
2^9|a^2=f,b^2=fghi,c^2=fgh,d^2=f,e^2=gh,b^a=bf,c^a=cg,d^a=di,e^a=efg,c^b=ch,d^b=dfg,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, 2680632);