Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7021363 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×D42 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2202 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 5 generators on 24 points:
(1,2)(3,5)(4,6)(7,8)(9,10,15,24,16,19,14,18)(11,20,21,12,22,13,17,23);(1,2)(3,7)(4,6)(5,8)(9,11)(10,12)(13,18)(14,21)(15,17)(16,22)(19,23)(20,24);(1,3)(2,5)(4,8)(6,7)(9,12,15,13,16,23,14,20)(10,17,18,22,19,21,24,11);(3,8)(5,7)(9,13,14,12,16,20,15,23)(10,17,18,22,19,21,24,11);(2,6)(5,7)(9,14,16,15)(10,18,19,24)(11,21,22,17)(12,13,23,20)Polycyclic presentation:
2^9|a^2=fi,c^2=g,d^2=f,e^2=i,f^2=i,g^2=i,b^a=bf,c^a=cg,e^a=eh,g^a=gi,c^b=ch,d^b=df,e^b=eh,f^b=fi,d^c=dh,e^c=ei,e^d=ei62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6928566);