Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6535632 |
| 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 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C42×D4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1614 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)(3,7)(4,6)(5,8)(9,10,15,14)(11,12,13,16)(17,18,22,24)(19,23,21,20);(1,3,5,4)(2,6,8,7)(9,11,15,13)(10,16,14,12)(17,19,22,21)(18,23,24,20);(9,12,15,16)(10,11,14,13)(17,20,21,18,22,23,19,24);(1,3,5,4)(2,7,8,6)(9,13,15,11)(10,16,14,12);(1,4,5,3)(2,6,8,7)(9,14,15,10)(11,12,13,16)Polycyclic presentation:
2^9|a^2=h,b^2=fi,c^2=g,d^2=f,e^2=f,g^2=i,b^a=bf,c^a=cg,g^a=gi,c^b=ch,e^d=ehi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6477357);