Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #93616 |
| automorphism | ? |
| homology | Abelianization (H1) = C42 · Schur multiplier (H2) = C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #1309 |
| commutator | C22×C8 | C42 |
| Frattini, cosocle | C42×C8 | C22 |
| socle | C2 | #7042 |
| prime radical | C23 | H(3,ℤ/4) |
| LCS | G ▷ C22×C8 ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C22×C4 ◁ C42×C8 ◁ G | |
| LECS | G ▷ C42×C8 ▷ C22×C4 ▷ C2 ▷ C1 | |
184 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 2 generators on 32 points:
(1,2)(3,12,14,27,15,23,16,28)(4,8)(5,11)(6,10)(7,13,20,29,22,30,21,31)(9,17)(18,26)(19,25)(24,32);(1,3)(2,7,9,21,10,22,11,20)(4,13)(5,14)(6,15)(8,23,24,27,25,12,26,28)(16,17)(18,29)(19,30)(31,32)Polycyclic presentation:
2^9|a^2=d,b^2=e,c^2=fhi,d^2=f,e^2=g,h^2=i,b^a=bc,c^a=cf,e^a=efghi,g^a=gi,c^b=cg,d^b=dh,f^b=fi,h^b=hi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 812);