Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #499724 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C22×D8 |
| commutator, Frattini, cosocle | C22×C8 | C24 |
| prime radical | C2×#3283 | C2 |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×C4×D4 ◁ C2×C8×D4 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
1083 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 4 generators on 40 points:
(2,3)(5,6)(7,8)(9,10,14,18)(11,19,21,38)(12,23,27,37)(13,28,26,17)(15,30,31,20)(16,33,35,24)(22,39,29,34)(25,36,40,32);(1,2,4,3)(11,20)(12,24)(13,26)(15,19)(16,23)(17,28)(21,30)(22,40)(25,29)(27,33)(31,38)(32,39)(34,36)(35,37);(6,8)(9,11,14,21)(10,15,18,31)(12,25,27,40)(13,29,26,22)(16,34,35,39)(17,36,28,32)(19,37,38,23)(20,33,30,24);(1,3,4,2)(11,21)(15,31)(19,38)(20,30)(22,29)(25,40)(32,36)(34,39)Polycyclic presentation:
2^9|a^2=i,b^2=e,c^2=i,d^2=e,f^2=hi,g^2=hi,h^2=i,b^a=be,c^a=cf,d^a=de,f^a=fh,g^a=gh,h^a=hi,c^b=cg,f^b=fh,g^b=gh,h^b=hi,d^c=di,f^c=fh,g^c=gh,h^c=hi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 407901);