Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6533105 |
| 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 | #24440 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C42×D4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1938 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 24 points:
(1,2,5,7)(3,8,4,6)(17,18,19,23,20,22,24,21);(1,3,5,4)(2,6,7,8)(9,10)(11,14)(12,13)(15,16);(10,13)(14,16)(18,21)(19,24)(22,23);(1,4,5,3)(2,6,7,8)(9,11)(10,14)(12,15)(13,16);(11,15)(14,16)(18,22)(21,23)Polycyclic presentation:
2^9|a^2=fg,b^2=f,d^2=f,g^2=i,b^a=bf,c^a=cg,e^a=ei,c^b=ch,g^c=gi,e^d=eh77 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6455325);