Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #13316 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #447 |
| commutator, Frattini, cosocle | C22×C8 | C24 |
| socle | C22 | C2×#447 |
| prime radical | #60676 | C2 |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C2×C42 ◁ C2×C4×M16 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
854 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)(4,5)(7,8)(9,10)(11,19)(12,23)(13,28)(14,17)(15,20)(16,27)(18,26)(21,39)(22,36)(24,35)(25,34)(29,33)(30,37)(31,40)(32,38);(2,5)(6,8)(12,24)(15,30)(16,32)(20,37)(23,35)(25,40)(27,38)(31,34);(9,11,14,22)(10,15,17,30)(12,25,24,40)(13,29,26,21)(16,33,32,39)(18,34,28,31)(19,35,36,23)(20,38,37,27);(1,3,4,7)(2,6,5,8)Polycyclic presentation:
2^9|c^2=i,d^2=e,f^2=hi,h^2=i,b^a=be,c^a=cf,f^a=fh,g^a=gi,h^a=hi,c^b=cg,f^b=fi,f^c=fh,h^c=hi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 406438);