Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #326959 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C2×#120 |
| commutator, Frattini, cosocle | C2×C42 | C24 |
| prime radical | #2541 | C22 |
| LCS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C43 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
1023 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 4 generators on 24 points:
(2,3)(6,7)(9,10)(11,13)(12,15)(14,16)(19,24)(21,23);(1,2)(3,5)(4,6)(7,8)(17,18,20,22)(19,21,24,23);(1,3,5,2)(4,6,8,7)(10,13)(11,16)(14,15)(18,21)(19,24)(22,23);(1,4,5,8)(2,6,3,7)Polycyclic presentation:
2^9|b^2=i,c^2=e,d^2=e,f^2=h,g^2=i,b^a=be,c^a=cf,f^a=fh,g^a=gi,c^b=cg,f^b=fi,g^b=gi,d^c=de,f^c=fh,g^c=gi65 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 236260);