Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #512609 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C25×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | C22×#353 |
| commutator | C42 | C23×C4 |
| Frattini, cosocle | #124 | C24 |
| prime radical | #60650 | C2 |
| LCS | G ▷ C42 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C2×D4 ◁ C2×#41 ◁ #3500 ◁ G | |
| LECS | G ▷ #124 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
1114 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2,4,10)(3,8,13,23)(5,9,15,24)(6,11,16,25)(7,17,20,18)(12,21,26,31)(14,22,28,32)(19,29,30,27);(2,7)(4,15)(5,18)(8,19)(9,11)(13,28)(14,27)(16,17)(20,24)(21,22)(26,29)(30,32);(1,3)(2,8)(4,13)(5,14)(6,12)(7,19)(9,22)(10,23)(11,21)(15,28)(16,26)(17,29)(18,27)(20,30)(24,32)(25,31);(3,12)(8,21)(13,26)(14,27)(19,22)(23,31)(28,29)(30,32)Polycyclic presentation:
2^9|a^2=f,e^2=h,g^2=i,b^a=be,e^a=eg,g^a=gh,h^a=hi,e^b=eh,f^b=fghi,g^b=gi,d^c=di,f^e=fhi,g^f=gi53 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 419805);