Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6682628 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×D42 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | #61217 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1797 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 40 points:
(1,2,5,7)(3,8,4,6)(9,10)(11,17)(12,24)(13,19)(14,30)(15,20)(16,21)(18,23)(22,32)(25,40)(26,33)(27,35)(28,37)(29,38)(31,36)(34,39);(1,2)(3,6)(4,8)(5,7)(9,11,16,23)(10,17,21,18)(12,15,26,14)(13,27,29,25)(19,35,38,40)(20,33,30,24)(22,31,39,28)(32,36,34,37);(9,12,14,23,16,26,15,11)(10,18,20,24,21,17,30,33)(13,25,28,39,29,27,31,22)(19,34,36,40,38,32,37,35);(1,3,5,4)(2,6,7,8);(1,4,5,3)(2,6,7,8)(9,13,16,29)(10,19,21,38)(11,22,23,39)(12,25,26,27)(14,28,15,31)(17,32,18,34)(20,36,30,37)(24,40,33,35)Polycyclic presentation:
2^9|a^2=f,b^2=i,c^2=g,d^2=f,e^2=fi,g^2=i,h^2=i,b^a=bf,c^a=cg,d^a=df,g^a=gi,c^b=ch,e^b=efh,h^b=hi,h^c=hi,h^e=hi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6594175);