Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #2230 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C8 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C8 | #476 |
| commutator | C2×C8 | C22×C8 |
| Frattini, cosocle | C2×C4×C8 | C23 |
| socle | C2 | #8563 |
| prime radical | C2×#135 | C2×C4 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C8 ◁ C22×C8 ◁ C42×C8 ◁ G | |
| LECS | G ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
392 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 3 generators on 32 points:
(1,2,6,12,7,11,17,29,8,13,16,9,5,24,3,14)(4,18,21,10,22,25,15,26,23,31,20,28,19,30,32,27);(1,3,5,16,8,17,7,6)(2,9,11,14,13,12,24,29)(4,15,19,21,23,32,22,20)(10,25,26,31,28,30,27,18);(1,4)(2,10)(3,15)(5,19)(6,20)(7,22)(8,23)(9,25)(11,26)(12,30)(13,28)(14,31)(16,21)(17,32)(18,29)(24,27)Polycyclic presentation:
2^9|a^2=f,b^2=d,d^2=i,e^2=g,f^2=h,g^2=i,h^2=i,b^a=bd,c^a=ce,d^a=di,e^a=eg,g^a=gi,e^c=egi,f^c=fi,g^c=gi92 conjugacy classes — view the full character table.
| maximal subgroup of | #72225414, #72225418, #72225419, … |
| maximal normal subgroup of | #72225414, #72225418, #72225419, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 41569);