Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #298603 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C23×C8 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C2×#127 |
| commutator, Frattini, cosocle | C2×C42 | C24 |
| prime radical | #28437 | C2 |
| LCS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C43 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
988 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 4 generators on 24 points:
(1,2)(3,7)(4,5)(6,8)(9,10,14,15,13,11,12,16);(2,5)(6,7)(10,11)(15,16)(17,18,19,22,20,23,24,21);(9,11,13,10)(12,16,14,15)(18,21)(19,24)(22,23);(1,3,4,8)(2,6,5,7)(9,12,13,14)(10,15,11,16)Polycyclic presentation:
2^9|a^2=fh,b^2=gh,c^2=h,d^2=e,f^2=h,g^2=i,b^a=be,c^a=cf,d^a=de,c^b=cg,f^c=fh,g^c=gi98 conjugacy classes — view the full character table.
| maximal subgroup of | #72224717, #72224719, #72224726 |
| maximal normal subgroup of | #72224717, #72224719, #72224726 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 227328);