Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #499855 |
| 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 | |
821 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)(10,16)(11,21)(12,15)(14,26)(17,32)(22,40)(23,39)(24,27)(28,38)(30,33)(34,36)(35,37);(1,3,5,4)(2,6,8,7)(9,10,13,16)(11,22,24,35)(12,20,15,18)(14,19,26,31)(17,28,30,23)(21,37,27,40)(25,36,29,34)(32,39,33,38);(9,11)(10,17)(12,23)(13,24)(14,27)(15,28)(16,30)(18,22)(19,32)(20,35)(21,26)(25,39)(29,38)(31,33)(34,37)(36,40);(1,4,5,3)(2,7,8,6)(9,12,13,15)(10,18,16,20)(11,23,24,28)(14,25,26,29)(17,22,30,35)(19,34,31,36)(21,38,27,39)(32,37,33,40)Polycyclic presentation:
2^9|a^2=ei,b^2=e,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,d^b=di,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, 408175);