Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #24404 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C29 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C2×#120 |
| commutator, Frattini, cosocle | C24 | C25 |
| socle | C22 | C22×#120 |
| prime radical | #62773 | C2 |
| LCS | G ▷ C24 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C24 ▷ C2 ▷ C1 | |
3421 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)(3,6)(4,5)(7,8)(9,10,13,16)(11,14,18,22)(12,15,20,23)(17,21,19,24);(9,11)(10,14)(12,19,20,17)(13,18)(15,21,23,24)(16,22);(2,5)(6,8)(9,12)(10,15)(11,17)(13,20)(14,21)(16,23)(18,19)(22,24);(9,12)(10,15)(11,17)(13,20)(14,21)(16,23)(18,19)(22,24);(1,3,4,7)(2,6,5,8)Polycyclic presentation:
2^9|a^2=i,b^2=f,e^2=g,b^a=bf,c^a=cg,h^a=hi,c^b=ch,d^b=dh,h^b=hi,f^c=fi,f^d=fi86 conjugacy classes — view the full character table.
| maximal subgroup of | #72285362, #72285473, #72316256, … |
| maximal normal subgroup of | #72285362, #72285473, #72316256, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6362066);