Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #501120 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×B(4,𝔽2) |
| commutator, Frattini, cosocle | C22×Q8 | C24 |
| prime radical | C22×D4 | C2×D4 |
| LCS | G ▷ C22×Q8 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ C2×C4×Q8 ◁ G | |
| LECS | G ▷ C22×Q8 ▷ C22 ▷ C2 ▷ C1 | |
746 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 40 points:
(1,2,4,5)(3,7,8,6)(9,10,13,18)(11,19,14,16)(12,23,25,31)(15,30,28,20)(17,21,34,26)(22,32,29,36)(24,39,40,35)(27,38,37,33);(2,5)(6,7)(9,11,13,14)(10,16,18,19)(12,24,25,40)(15,29,28,22)(17,33,34,38)(20,36,30,32)(21,37,26,27)(23,35,31,39);(9,12,14,26,13,25,11,21)(10,17,19,23,18,34,16,31)(15,27,22,24,28,37,29,40)(20,35,32,33,30,39,36,38);(1,3,4,8)(2,6,5,7)(10,18)(16,19)(17,34)(20,30)(23,31)(32,36)(33,38)(35,39)Polycyclic presentation:
2^9|a^2=e,b^2=i,c^2=f,d^2=ei,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=de,f^a=fi,g^a=ghi,c^b=cg,f^b=fh,g^b=gi,g^c=ghi,h^c=hi,g^f=gi47 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 410191);