Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7519083 |
| 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 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1815 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 40 points:
(1,2,5,6)(3,7,4,8)(9,10,14,19)(11,21,24,16)(12,13,27,29)(15,31,28,20)(17,35,30,18)(22,39,33,36)(23,26,38,40)(25,34,32,37);(2,6)(7,8)(9,11,14,24)(10,16,19,21)(12,26,27,40)(13,23,29,38)(15,32,28,25)(17,36,30,39)(18,22,35,33)(20,37,31,34);(1,3,5,4)(2,7,6,8)(9,12,15,18,14,27,28,35)(10,17,20,13,19,30,31,29)(11,22,25,26,24,33,32,40)(16,23,34,36,21,38,37,39);(9,13,28,17,14,29,15,30)(10,18,31,12,19,35,20,27)(11,23,32,39,24,38,25,36)(16,33,37,26,21,22,34,40);(1,4,5,3)(2,7,6,8)(10,19)(12,27)(16,21)(18,35)(20,31)(22,33)(26,40)(34,37)Polycyclic presentation:
2^9|a^2=f,b^2=i,c^2=fgi,d^2=gi,e^2=fi,g^2=i,b^a=bf,c^a=cg,d^a=dg,e^a=ef,g^a=gi,h^a=hi,c^b=cg,d^b=dh,g^b=gi,e^c=ei,h^d=hi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7472021);