Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6533087 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | #36444 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×C4×#41 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1921 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 24 points:
(1,2,5,7)(3,8,4,6)(9,10,14,13)(11,16,12,15)(17,18,19,22,20,23,24,21);(1,3,5,4)(2,6,7,8)(9,11,14,12)(10,15,13,16)(18,21)(19,24)(22,23);(1,4,5,3)(2,6,7,8)(9,10,14,13)(11,16,12,15)(18,21)(19,24)(22,23);(9,12,14,11)(10,15,13,16);(9,13,14,10)(11,16,12,15)Polycyclic presentation:
2^9|a^2=f,b^2=fgi,c^2=fgi,d^2=h,e^2=h,f^2=i,g^2=i,b^a=bf,c^a=cg,c^b=ch,f^b=fi,g^b=gi,f^c=fi,g^c=gi,e^d=eh89 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6470471);