Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #510363 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C22×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #31547 |
| commutator, Frattini, cosocle | C22×C8 | C24 |
| prime radical | C2×#509 | C22 |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C22×C4 ◁ #1773 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
429 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2,7,13)(3,14)(4,6,12,27)(5,9,15,11)(8,19)(10,24,26,23)(16,28)(17,25)(18,30)(20,22)(21,29)(31,32);(1,3,15,19,7,21,5,18)(2,8,9,14,13,30,11,29)(4,22,23,28,12,31,24,17)(6,25,26,32,27,16,10,20);(2,9)(3,16)(5,15)(6,26)(8,28)(10,27)(11,13)(14,22)(17,30)(18,20)(19,32)(21,25)(23,24)(29,31);(1,4,7,12)(2,10,13,26)(3,17)(5,23,15,24)(6,9,27,11)(8,25)(14,20)(16,30)(18,22)(19,31)(21,28)(29,32)Polycyclic presentation:
2^9|a^2=eh,b^2=efh,d^2=eh,e^2=i,f^2=i,g^2=h,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=ghi,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, 417559);