Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6535648 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | #36127 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C42×D4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1802 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 24 points:
(1,2,5,6)(3,7,4,8)(9,10,13,11)(12,14,16,15)(17,18,21,23)(19,24,22,20);(2,6)(7,8)(9,11,13,10)(12,15,16,14)(17,18,20,22,21,23,24,19);(1,3,5,4)(2,7,6,8)(10,11)(14,15)(17,19,20,18,21,22,24,23);(1,4,5,3)(2,8,6,7)(12,16)(14,15)(17,20,21,24)(18,19,23,22);(9,12,13,16)(10,14,11,15)Polycyclic presentation:
2^9|a^2=fg,b^2=g,c^2=f,d^2=fghi,e^2=hi,f^2=i,g^2=i,b^a=bf,c^a=cg,f^a=fi,g^a=gi,c^b=ch,d^b=di,d^c=di,e^d=ehi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6465026);