Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #190432 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C25×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×#381 |
| commutator, prime radical | C24 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| LCS | G ▷ C24 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1189 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 40 points:
(1,2)(3,7)(4,6)(5,8)(9,10,13,18,14,15,26,28)(11,19,22,35,21,16,29,31)(12,23,24,33,25,17,27,32)(20,30,37,39,36,34,38,40);(1,3,4,8)(2,5,6,7)(10,15)(13,26)(16,19)(17,23)(22,29)(24,27)(30,34)(37,38);(3,8)(5,7)(9,11)(10,16,15,19)(12,20)(13,22)(14,21)(17,30,23,34)(18,31,28,35)(24,37)(25,36)(26,29)(27,38)(32,39,33,40);(2,6)(5,7)(9,12,14,25)(10,17,15,23)(11,20,21,36)(13,27,26,24)(16,30,19,34)(18,33,28,32)(22,38,29,37)(31,40,35,39)Polycyclic presentation:
2^9|a^2=h,b^2=g,c^2=eg,d^2=i,h^2=i,b^a=be,c^a=cf,d^a=de,e^a=ei,c^b=cg,h^b=hi,f^c=fi,h^d=hi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 270064);