Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #188774 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C2×#117 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C22×C8 | C24 |
| socle | C22 | C2×#432 |
| prime radical | C23 | C22×M16 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C4×#97 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
422 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)(3,6)(4,8)(5,7)(9,10,14,19,13,20,30,37,15,21,32,17,31,18,12,28)(11,22,25,33,26,35,40,36,27,38,23,39,24,16,29,34);(1,3,5,4)(2,6,7,8)(9,11,14,25,13,26,30,40,15,27,32,23,31,24,12,29)(10,16,19,34,20,22,37,33,21,35,17,36,18,38,28,39);(1,4,5,3)(2,6,7,8);(9,12,13,14,15,30,31,32)(10,17,18,37,21,19,20,28)(11,23,24,40,27,25,26,29)(16,33,22,36,35,39,38,34)Polycyclic presentation:
2^9|a^2=g,b^2=fg,c^2=f,d^2=e,e^2=i,g^2=h,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,d^b=dh,e^b=ei,g^d=gi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 402639);