Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #499867 |
| 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 | C2×2−1+4 | C23 |
| LCS | G ▷ C22×Q8 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ C2×C4×Q8 ◁ G | |
| LECS | G ▷ C22×Q8 ▷ C22 ▷ C2 ▷ C1 | |
825 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 40 points:
(1,2,5,8)(3,7,4,6)(9,10,13,16)(11,21,24,35)(12,19,14,18)(15,29,26,20)(17,27,31,23)(22,38,30,37)(25,34,28,36)(32,39,33,40);(1,3,5,4)(2,6,8,7)(10,16)(11,22)(15,26)(17,32)(18,19)(21,37)(23,39)(24,30)(25,28)(27,40)(31,33)(35,38);(9,11)(10,17)(12,23)(13,24)(14,27)(15,30)(16,31)(18,21)(19,35)(20,32)(22,26)(25,40)(28,39)(29,33)(34,37)(36,38);(1,4,5,3)(2,7,8,6)(9,12,13,14)(10,18,16,19)(11,23,24,27)(15,25,26,28)(17,21,31,35)(20,34,29,36)(22,39,30,40)(32,37,33,38)Polycyclic presentation:
2^9|a^2=e,b^2=ei,d^2=e,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,f^c=fi,g^c=gi,h^c=hi,g^f=gi50 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 408166);