Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6533087 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C26×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | #36258 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×C4×#41 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1929 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)(3,8)(4,6)(5,7)(18,19)(20,21)(23,24);(1,3,5,4)(2,6,7,8)(9,10)(11,14)(12,13)(15,16)(17,18,22,24)(19,20,23,21);(1,4,5,3)(2,6,7,8)(10,13)(14,16)(17,19)(18,20)(21,24)(22,23);(9,11)(10,14)(12,15)(13,16);(11,15)(14,16)Polycyclic presentation:
2^9|b^2=fgi,c^2=fg,f^2=i,g^2=i,b^a=bf,c^a=cg,f^a=fi,g^a=gi,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, 6468976);