Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6437963 |
| 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 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
3406 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)(3,8)(4,6)(5,7)(9,10,15,18)(11,13,17,14)(12,16,21,23)(19,20,24,22);(9,11,14,10,15,17,13,18)(12,19,20,23,21,24,22,16);(1,3,5,4)(2,6,7,8)(9,12)(10,16)(11,19)(13,20)(14,22)(15,21)(17,24)(18,23);(1,4,5,3)(2,6,7,8)(9,12)(10,16)(11,19)(13,20)(14,22)(15,21)(17,24)(18,23);(9,13,15,14)(10,11,18,17)(12,20,21,22)(16,19,23,24)Polycyclic presentation:
2^9|a^2=i,b^2=f,c^2=g,d^2=g,e^2=i,f^2=i,b^a=bf,c^a=cg,e^a=ei,f^a=fi,h^a=hi,c^b=ch,d^b=dh,h^b=hi,f^c=fi,f^d=fi80 conjugacy classes — view the full character table.
| maximal subgroup of | #72252472, #72252473, #72252485, … |
| maximal normal subgroup of | #72252472, #72252473, #72252485, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6359448);