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 | |
2199 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)(9,10,15,19,16,21,14,20)(11,13,18,17,23,24,22,12);(1,2)(3,7)(4,6)(5,8)(9,11)(10,17)(12,21)(13,19)(14,18)(15,22)(16,23)(20,24);(1,3)(2,5)(4,8)(6,7)(9,12,14,24,16,17,15,13)(10,18,19,23,21,22,20,11);(3,8)(5,7)(9,13,14,12,16,24,15,17)(10,18,20,11,21,22,19,23);(2,6)(5,7)(9,14,16,15)(10,19,21,20)(11,22,23,18)(12,13,17,24)Polycyclic presentation:
2^9|a^2=fi,c^2=gi,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, 6928568);