Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #93313 |
| automorphism | ? |
| homology | Abelianization (H1) = C4×C8 · Schur multiplier (H2) = C2×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #1281 |
| commutator | C2×C8 | C4×C8 |
| Frattini, cosocle | C4×#97 | C22 |
| socle | C2 | #7019 |
| prime radical | C23 | H(3,ℤ/4) |
| LCS | G ▷ C2×C8 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C22×C8 ◁ C4×#97 ◁ G | |
| LECS | G ▷ C4×#97 ▷ C22×C4 ▷ C2 ▷ C1 | |
175 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)(3,12,14,27,17,28,16,31)(4,18,20,9)(5,21,22,32)(6,23)(7,10)(8,13,24,15,25,29,26,30)(11,19);(1,3,5,15,6,16,20,29,7,17,22,30,19,14,4,13)(2,8,9,12,11,26,21,27,10,25,18,28,23,24,32,31)Polycyclic presentation:
2^9|a^2=d,b^2=e,c^2=fgi,d^2=h,e^2=g,g^2=i,b^a=bc,c^a=cf,e^a=efgi,f^a=fi,g^a=gi,d^b=dg,h^b=hi,d^c=di,e^d=ei68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 517);