Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #511495 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C22×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #32613 |
| commutator, Frattini, cosocle | C22×C8 | C24 |
| prime radical | #13331 | C2 |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C22×C4 ◁ #2960 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
452 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2,8,4,6,12,5,11)(3,13,16,9,18,19,15,26)(7,23,25,29,22,10,21,20)(14,30,31,32,27,17,28,24);(1,3)(2,9)(4,19)(5,15)(6,18)(7,24)(8,16)(10,28)(11,13)(12,26)(14,20)(17,21)(22,32)(23,31)(25,30)(27,29);(1,4,5,12,6,11,8,2)(3,14,16,28,18,27,15,31)(7,23,21,29,22,10,25,20)(9,24,13,30,26,32,19,17);(3,15,18,16)(9,19,26,13)(14,31,27,28)(17,32,30,24)Polycyclic presentation:
2^9|a^2=ef,c^2=e,d^2=efh,e^2=i,g^2=hi,h^2=i,b^a=be,c^a=cf,d^a=defh,e^a=ei,f^a=fi,g^a=gi,c^b=cg,d^b=de,e^b=ei,f^b=fi,g^b=gh,h^b=hi,f^c=fi,g^c=ghi,h^c=hi44 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 418691);