Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6438991 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C29 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | #35565 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×#525 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2945 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 24 points:
(2,4)(9,10,16,13,15,11,14,18)(12,17,23,20,21,19,22,24);(5,6)(7,8)(9,11,15,10)(12,20,21,24)(13,14,18,16)(17,22,19,23);(1,2,3,4)(6,8)(9,12,14,22,15,21,16,23)(10,17,18,24,11,19,13,20);(6,8)(9,13,14,10,15,18,16,11)(12,20,22,17,21,24,23,19);(12,21)(17,19)(20,24)(22,23)Polycyclic presentation:
2^9|a^2=fi,b^2=i,c^2=fg,d^2=f,f^2=i,h^2=i,b^a=bf,c^a=cg,c^b=ch,d^b=dh,f^b=fi,h^b=hi,e^c=ei92 conjugacy classes — view the full character table.
| maximal subgroup of | #72315219, #72315222, #72315231 |
| maximal normal subgroup of | #72315219, #72315222, #72315231 |
| maximal quotient of | #72315685, #72315742 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6351563);