Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #477926 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C23×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #17126 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C2×#535 | C22 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
881 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2,6,12,8,11,7,3)(4,13,17,9,19,27,18,15)(5,20,21,26,23,14,22,10)(16,25,30,28,31,24,32,29);(1,3,6,11,8,12,7,2)(4,15,17,27,19,9,18,13)(5,10,21,14,23,26,22,20)(16,28,30,25,31,29,32,24);(1,4,6,17,8,19,7,18)(2,9,3,13,11,15,12,27)(5,16,21,30,23,31,22,32)(10,25,14,29,26,24,20,28);(1,5)(2,10,11,26)(3,14,12,20)(4,16)(6,21)(7,22)(8,23)(9,24,15,25)(13,28,27,29)(17,30)(18,32)(19,31)Polycyclic presentation:
2^9|a^2=ef,b^2=f,c^2=f,d^2=e,f^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,h^a=hi,c^b=cg,d^b=def,e^b=ei,g^b=gi,d^c=de,g^c=gi,h^c=hi,g^d=gi47 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 385122);