Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #4462490 |
| automorphism | Aut(G) order 2097152 |
| 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 | |
1398 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 5 generators on 40 points:
(1,2,4,5)(3,7,8,6)(10,13)(14,16)(19,23)(21,24)(25,26)(27,32)(28,29)(30,31)(33,37)(34,40)(35,38)(36,39);(2,5)(6,7)(11,15)(14,16)(17,18,20,22)(19,21,23,24)(26,31)(28,37)(32,39)(34,38);(1,2,4,5)(3,6,8,7)(9,10)(11,14)(12,13)(15,16)(25,27,30,36)(26,32,31,39)(28,38,37,34)(29,35,33,40);(1,2)(3,6)(4,5)(7,8)(9,11,12,15)(10,14,13,16)(17,19)(18,21)(20,23)(22,24)(25,28)(26,33)(27,34)(29,31)(30,37)(32,35)(36,38)(39,40);(1,3)(2,6)(4,8)(5,7)(25,29,30,33)(26,28,31,37)(27,35,36,40)(32,38,39,34)Polycyclic presentation:
2^9|a^2=h,b^2=fi,c^2=f,d^2=g,e^2=fh,b^a=bf,c^a=cg,d^a=di,e^a=eh,c^b=ch,d^b=dfg,d^c=df68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 4371185);