Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #501072 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×#447 |
| commutator, Frattini, cosocle | C22×C8 | C24 |
| prime radical | C2×#41 | C2×D4 |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ C4×#131 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
724 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 4 generators on 40 points:
(1,2,5,8)(3,7,4,6)(9,10)(11,17)(12,25)(13,29)(14,19)(15,32)(16,20)(18,27)(21,37)(22,39)(23,34)(24,36)(26,40)(28,30)(31,35)(33,38);(1,3,5,4)(2,6,8,7)(9,11,16,24)(10,17,20,36)(12,26,18,33)(13,22,30,37)(14,31,15,23)(19,35,32,34)(21,29,39,28)(25,40,27,38);(9,12,27,20,15,28,30,32,16,18,25,10,14,29,13,19)(11,21,38,34,23,33,37,17,24,39,40,35,31,26,22,36);(1,4,5,3)(2,7,8,6)Polycyclic presentation:
2^9|a^2=e,b^2=ei,c^2=fh,d^2=e,f^2=hi,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=de,f^a=fh,h^a=hi,c^b=cg,g^b=gi,g^c=gi74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 410494);