Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #446220 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C22×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #15835 |
| 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 | |
698 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2,5,10,6,12,19,11)(3,9,15,7,16,23,14,24)(4,17,20,8,21,27,32,28)(13,22,18,25,30,31,29,26);(1,3,5,14,6,16,19,15)(2,7,11,23,12,24,10,9)(4,18,20,13,21,29,32,30)(8,25,17,31,28,26,27,22);(1,4,19,32,6,21,5,20)(2,8,10,27,12,28,11,17)(3,13,14,29,16,30,15,18)(7,22,23,25,24,31,9,26);(1,3)(2,9,12,23)(4,13)(5,15)(6,16)(7,11,24,10)(8,26,28,25)(14,19)(17,31,27,22)(18,20)(21,30)(29,32)Polycyclic presentation:
2^9|a^2=g,b^2=f,c^2=efg,d^2=efi,e^2=i,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=dh,f^a=fi,c^b=cg,d^b=de,g^b=gi,h^b=hi,h^c=hi,e^d=ei,f^d=fi,h^d=hi59 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 353417);