Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #510293 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #31483 |
| commutator, Frattini, cosocle | C22×C8 | C24 |
| prime radical | C22×D8 | C23 |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C22×C4 ◁ #1773 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
362 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2,8,14)(3,15)(4,7,13,26)(5,12,6,10)(9,20)(11,23,24,25)(16,17)(18,30)(19,22)(21,29)(27,28)(31,32);(1,3)(2,9,14,30)(4,22,13,32)(5,18)(6,20)(7,16)(8,21)(10,29,12,15)(11,31)(17,23,27,25)(19,24)(26,28);(2,10)(3,16)(5,6)(7,24)(9,27)(11,26)(12,14)(15,22)(17,30)(18,31)(19,20)(21,28)(23,25)(29,32);(1,4,8,13)(2,11,14,24)(3,17)(5,23,6,25)(7,10,26,12)(9,28)(15,31)(16,30)(18,22)(19,29)(20,32)(21,27)Polycyclic presentation:
2^9|a^2=eh,b^2=fh,d^2=eh,e^2=i,f^2=i,g^2=hi,h^2=i,b^a=be,c^a=cf,d^a=defh,f^a=fi,c^b=cg,d^b=de,f^b=fi,g^b=gh,h^b=hi,e^c=ei,f^c=fi,g^c=gh,h^c=hi,f^d=fi,g^d=gi41 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 417489);