Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6996639 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | C2×#496 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2399 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1273 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)(4,5)(7,8)(12,20)(15,21)(19,24)(22,23)(25,26,30,34)(27,33,31,28)(29,32,38,40)(35,39,37,36);(2,5)(3,7)(9,10,13,16)(11,17,18,14)(12,21,20,15)(19,22,24,23)(25,27,30,31)(26,28,34,33)(29,37,38,35)(32,39,40,36);(9,11)(10,14)(12,19)(13,18)(15,22)(16,17)(20,24)(21,23)(25,28,31,26,30,33,27,34)(29,36,37,32,38,39,35,40);(1,3,4,7)(2,6,5,8)(11,18)(14,17)(19,24)(22,23)(25,29)(26,32)(27,35)(28,36)(30,38)(31,37)(33,39)(34,40);(9,12,13,20)(10,15,16,21)(11,19,18,24)(14,22,17,23)(25,27,30,31)(26,33,34,28)(29,37,38,35)(32,36,40,39)Polycyclic presentation:
2^9|a^2=i,b^2=hi,c^2=g,d^2=fi,e^2=hi,g^2=i,b^a=bf,c^a=cg,e^a=eh,g^a=gi,c^b=ch,d^b=df,e^b=eh,d^c=dh,e^c=ei,e^d=ei68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7093652);