Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6502828 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C22×#121 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | #60916 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1746 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 24 points:
(9,10,13,16,14,18,23,17)(11,19,22,24,21,12,20,15);(1,2)(3,6)(4,5)(7,8)(9,11,14,21)(10,15,18,24)(12,17,19,16)(13,20,23,22);(2,5)(3,7)(9,12,23,24,14,19,13,15)(10,11,16,22,18,21,17,20);(1,3,4,7)(2,6,5,8)(9,13,14,23)(10,16,18,17)(11,20,21,22)(12,24,19,15);(2,5)(6,8)(11,21)(12,19)(15,24)(20,22)Polycyclic presentation:
2^9|a^2=f,b^2=i,c^2=gi,d^2=h,f^2=i,g^2=i,b^a=bf,c^a=cg,g^a=gi,c^b=ch,e^b=eh,f^b=fi,d^c=dh,e^c=ei68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6427230);