Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of D4×#453 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C29 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C22×#120 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | C2×#454 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C4×Q8 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2835 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 24 points:
(9,10)(11,18)(12,15)(13,17)(14,22)(16,20)(19,23)(21,24);(1,2,6,4)(3,8,5,7)(9,11)(10,16)(12,19)(13,20)(14,23)(15,21)(17,24)(18,22);(1,3,6,5)(2,7,4,8)(9,12,15,10)(11,19,21,16)(13,22,14,17)(18,23,24,20);(1,4,6,2)(3,7,5,8)(9,13,15,14)(10,17,12,22)(11,20,21,23)(16,24,19,18);(1,5,6,3)(2,8,4,7)Polycyclic presentation:
2^9|b^2=h,c^2=hi,d^2=hi,e^2=h,f^2=i,b^a=bf,c^a=cg,f^a=fi,c^b=ch,e^b=eh,f^b=fi,g^b=gi,d^c=dh,f^c=fi,e^d=eh80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6532769);