Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #10792 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C4×C8 · Schur multiplier (H2) = C23×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C4×C22⋊C4 |
| commutator | C2×C4 | C2×C4×C8 |
| Frattini, cosocle | C22×C42 | C23 |
| socle | C22 | #1434 |
| prime radical | C24 | C4×D4 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C2×#431 ◁ G | |
| LECS | G ▷ C22×C42 ▷ C22 ▷ C1 | |
554 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 3 generators on 40 points:
(1,2,4,7,6,8,5,3)(9,10)(11,21,24,31)(12,26)(13,18)(14,19)(15,20)(16,23,32,25)(17,28)(22,36,39,30)(27,33)(29,34)(35,37,40,38);(1,3,5,8,6,7,4,2)(9,11,14,25,15,24,13,23)(10,16,18,21,20,32,19,31)(12,22,28,38,29,39,27,37)(17,30,26,35,33,36,34,40);(9,12,13,27,15,29,14,28)(10,17,18,26,20,33,19,34)(11,22,23,37,24,39,25,38)(16,30,21,35,32,36,31,40)Polycyclic presentation:
2^9|a^2=f,b^2=g,c^2=d,d^2=h,e^2=h,f^2=i,g^2=hi,b^a=bd,c^a=ce,e^a=eh,g^a=gh,f^b=fh,e^c=eh116 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 15632);