Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #2735949 |
| 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 | |
1432 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 5 generators on 40 points:
(1,2,5,6)(3,4,7,8)(9,10,13,14)(11,15,12,16)(25,26,30,34)(27,28,32,31)(29,37,39,33)(35,38,40,36);(2,6)(3,7)(10,14)(15,16)(18,21)(22,24)(25,27,30,32)(26,31,34,28)(29,35,39,40)(33,38,37,36);(1,3,5,7)(2,4,6,8)(9,11,13,12)(10,15,14,16)(17,18,20,21)(19,22,23,24)(25,27)(26,31)(28,34)(29,35)(30,32)(33,38)(36,37)(39,40);(1,3)(2,4)(5,7)(6,8)(9,10)(11,16)(12,15)(13,14)(25,28,30,31)(26,32,34,27)(29,38,39,36)(33,35,37,40);(1,4)(2,7)(3,6)(5,8)(9,12,13,11)(10,16,14,15)(17,19,20,23)(18,22,21,24)(25,29,30,39)(26,33,34,37)(27,35,32,40)(28,36,31,38)Polycyclic presentation:
2^9|a^2=f,b^2=g,c^2=ghi,d^2=g,e^2=fgh,b^a=bf,c^a=cg,d^a=di,e^a=eg,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, 2652634);