Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #2230 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C8 · Schur multiplier (H2) = C22 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C8 | #476 |
| commutator | C2×C8 | C22×C8 |
| Frattini, cosocle | C2×C4×C8 | C23 |
| socle | C2 | #9602 |
| prime radical | C23 | #430 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C8 ◁ C22×C8 ◁ C42×C8 ◁ G | |
| LECS | G ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
226 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 3 generators on 32 points:
(1,2,6,12,7,11,22,25,8,13,21,10,5,23,4,20)(3,14,16,29,18,30,15,28,19,31,24,9,17,26,32,27);(1,3,4,15,5,17,21,16,8,19,22,32,7,18,6,24)(2,9,25,31,23,28,12,30,13,29,20,14,11,27,10,26);(1,4,5,21,8,22,7,6)(2,10,11,20,13,12,23,25)(3,15,17,16,19,32,18,24)(9,26,27,14,29,30,28,31)Polycyclic presentation:
2^9|a^2=f,b^2=fg,c^2=e,d^2=g,e^2=i,f^2=h,g^2=i,h^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=ei,g^a=gi,d^b=dg,f^b=fi,g^b=gi92 conjugacy classes — view the full character table.
| maximal subgroup of | #72225416, #72225418, #72225420, … |
| maximal normal subgroup of | #72225416, #72225418, #72225420, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 47589);