Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6573671 |
| 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 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C4×D4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
3032 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 5 generators on 24 points:
(9,10,16,20,15,11,14,18)(12,17,21,23,13,19,22,24);(1,2,4,5)(3,6,7,8)(9,11,15,10)(12,17,13,19)(14,18,16,20)(21,24,22,23);(2,5)(3,7)(9,12)(10,17)(11,19)(13,15)(14,21)(16,22)(18,23)(20,24);(1,3)(2,6)(4,7)(5,8)(9,13,15,12)(10,17,11,19)(14,21,16,22)(18,24,20,23);(2,5)(3,7)(9,14,15,16)(10,18,11,20)(12,21,13,22)(17,23,19,24)Polycyclic presentation:
2^9|a^2=fi,b^2=h,d^2=i,e^2=i,f^2=i,b^a=bf,c^a=cg,d^a=di,g^a=gi,c^b=ch,e^b=eh,f^b=fi,g^b=gi,d^c=dh,f^c=fi,e^d=eh74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6503365);