Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of D8×AGL(1,ℤ/8) |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×D42 |
| commutator, Frattini, cosocle | C42 | C25 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C42 ▷ C22 ▷ C1 | |
1627 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 32 points:
(2,7)(3,5)(8,15)(9,13)(10,14)(11,16)(17,18,22,25)(19,26,28,31)(20,24,21,29)(23,30,32,27);(1,2,6,9)(3,7,5,13)(4,11,12,15)(8,14,16,10);(1,3,6,5)(2,7,9,13)(4,10,12,14)(8,15,16,11)(17,19)(18,23)(20,27)(21,30)(22,28)(24,31)(25,32)(26,29);(1,4,6,12)(2,8,9,16)(3,10,5,14)(7,15,13,11);(2,9)(7,13)(8,16)(11,15)(17,20,22,21)(18,24,25,29)(19,27,28,30)(23,31,32,26)Polycyclic presentation:
2^9|a^2=i,b^2=h,c^2=h,d^2=h,e^2=i,f^2=h,g^2=i,b^a=bf,c^a=cg,f^a=fh,g^a=gi,d^b=df,e^b=eh,f^b=fh,g^c=gi,f^d=fh77 conjugacy classes — view the full character table.
| maximal subgroup of | #72306654, #72306661, #72306665, … |
| maximal normal subgroup of | #72306654, #72306661, #72306665, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6266025);