Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #479151 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C22×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #17188 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C23×D4 | C23 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
661 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2,6,21,7,11,20,10)(3,12,5,8,15,23,14,24)(4,16,17,9,18,32,29,27)(13,22,19,26,31,25,30,28);(1,3,6,5,7,15,20,14)(2,8,10,12,11,24,21,23)(4,13,17,19,18,31,29,30)(9,25,16,26,27,22,32,28);(1,4,6,17,7,18,20,29)(2,9,10,16,11,27,21,32)(3,13,5,19,15,31,14,30)(8,22,12,28,24,25,23,26);(1,5,20,15,7,14,6,3)(2,8,21,12,11,24,10,23)(4,13,29,19,18,31,17,30)(9,26,32,22,27,28,16,25)Polycyclic presentation:
2^9|a^2=eg,b^2=e,c^2=e,d^2=fi,e^2=i,f^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,g^a=gi,h^a=hi,c^b=cg,d^b=def,f^b=fi,h^b=hi,d^c=de,h^c=hi,e^d=ei,h^d=hi47 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 386348);