Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #4462490 |
| 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 | |
1431 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 40 points:
(2,5)(6,8)(9,10,14,12)(11,15,13,16)(17,18,21,20)(19,22,23,24)(25,26,30,29)(27,33,35,31)(28,36,37,32)(34,38,40,39);(9,11,14,13)(10,15,12,16)(18,20)(22,24)(28,37)(32,36)(34,40)(38,39);(9,12,14,10)(11,15,13,16)(17,19)(18,22)(20,24)(21,23)(25,27,30,35)(26,31,29,33)(28,34,37,40)(32,38,36,39);(1,2)(3,6)(4,5)(7,8)(9,11,14,13)(10,16,12,15)(17,20,21,18)(19,22,23,24)(25,28)(26,32)(27,34)(29,36)(30,37)(31,38)(33,39)(35,40);(1,3,4,7)(2,6,5,8)(9,13,14,11)(10,15,12,16)(25,29,30,26)(27,33,35,31)(28,36,37,32)(34,39,40,38)Polycyclic presentation:
2^9|a^2=fgh,b^2=h,c^2=gh,d^2=fh,e^2=i,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, 4372933);