Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #32888 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C2×#120 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| socle | C22 | C2×#379 |
| prime radical | C23×#41 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
3468 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 40 points:
(1,2)(3,6)(4,5)(7,8)(9,10)(11,20)(12,24,26,33)(13,27)(14,18)(15,19)(16,22)(17,25,34,29)(21,36,39,30)(23,31)(28,32)(35,38,40,37);(2,5)(6,8);(1,3,4,7)(2,6,5,8)(9,11)(10,16)(12,21)(13,28)(14,22)(15,23)(17,30)(18,31)(19,32)(20,27)(24,35)(25,38)(26,39)(29,37)(33,40)(34,36);(9,12)(10,17)(11,21)(13,25)(14,29)(15,26)(16,30)(18,33)(19,34)(20,35)(22,37)(23,39)(24,27)(28,38)(31,40)(32,36)Polycyclic presentation:
2^9|a^2=h,c^2=e,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=dg,f^a=fi,f^c=fi,g^d=gi,h^d=hi98 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 299458);