Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #11692 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42 · Schur multiplier (H2) = C2×C42 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C2×#99 |
| commutator | C22×C4 | C2×C42 |
| Frattini, cosocle | C2×#117 | C23 |
| socle | C22 | C2×#342 |
| prime radical | C22×#41 | D4 |
| LCS | G ▷ C22×C4 ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C2×C42 ◁ C2×C4×#41 ◁ G | |
| LECS | G ▷ C2×#117 ▷ C23 ▷ C2 ▷ C1 | |
665 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 3 generators on 40 points:
(1,2)(3,7)(4,6)(5,8)(9,10,13,18,14,16,26,31)(11,19,21,34,22,32,35,40)(12,17,24,33,25,30,37,39)(15,27,29,23,28,36,38,20);(2,6,8,7)(9,11)(10,15)(12,23,25,20)(13,21)(14,22)(16,28)(17,32,30,19)(18,29)(24,36,37,27)(26,35)(31,38)(33,40,39,34);(1,3,5,4)(2,7,8,6)(10,16)(15,28)(17,30)(18,31)(19,32)(29,38)(33,39)(34,40)Polycyclic presentation:
2^9|a^2=e,b^2=f,c^2=g,d^2=g,e^2=i,b^a=bd,c^a=ci,d^a=dg,f^a=fgh,d^b=dh,h^b=hi,f^d=fi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 34734);