Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6574100 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C22×#121 |
| 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 | |
1829 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)(11,13)(12,20)(14,19)(15,17)(16,23)(18,22)(21,24);(1,2,4,5)(3,6,7,8)(9,11,13,10,15,19,14,17)(12,18,21,16,23,24,22,20);(2,5)(3,7)(9,12,14,22,15,23,13,21)(10,16,11,18,17,20,19,24);(1,3)(2,6)(4,7)(5,8)(9,13,15,14)(10,11,17,19)(12,21,23,22)(16,18,20,24);(2,5)(3,7)Polycyclic presentation:
2^9|b^2=fhi,c^2=f,d^2=i,f^2=i,b^a=bf,c^a=cg,f^a=fi,c^b=ch,d^b=di,e^b=eh,d^c=dh,g^c=gi,e^d=eh74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6505710);