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 | |
1438 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 5 generators on 40 points:
(1,2,6,8)(3,5,7,4)(11,15)(14,16)(17,18,21,19)(20,23,24,22)(25,26)(27,35)(28,37)(29,38)(30,31)(32,36)(33,39)(34,40);(1,3,6,7)(2,4,8,5)(10,13)(14,16)(26,31)(27,36)(33,37)(34,38);(1,4,6,5)(2,3,8,7)(9,10,12,13)(11,14,15,16)(17,19,21,18)(20,23,24,22)(25,27)(26,32)(28,38)(29,33)(30,36)(31,35)(34,39)(37,40);(1,3)(2,4)(5,8)(6,7)(9,11,12,15)(10,14,13,16)(17,20,21,24)(18,22,19,23)(25,28)(26,33)(27,34)(29,35)(30,39)(31,37)(32,40)(36,38);(1,5,6,4)(2,7,8,3)(25,29,30,40)(26,34,31,38)(27,33,36,37)(28,32,39,35)Polycyclic presentation:
2^9|a^2=hi,b^2=fg,c^2=gi,d^2=fi,e^2=f,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, 2654483);