Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6362950 |
| 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 | |
1619 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 32 points:
(1,2,7,5)(3,10,13,16)(4,9,6,14)(8,12,15,11)(17,18,22,25)(19,23,21,29)(20,27,28,31)(24,30,32,26);(1,3,7,13)(2,8,5,15)(4,11,6,12)(9,16,14,10)(17,19,22,21)(18,23,25,29)(20,26,28,30)(24,31,32,27);(1,4,7,6)(2,9,5,14)(3,11,13,12)(8,16,15,10)(17,20,22,28)(18,24,25,32)(19,26,21,30)(23,31,29,27);(1,5,7,2)(3,10,13,16)(4,14,6,9)(8,12,15,11);(1,4,7,6)(2,9,5,14)(3,12,13,11)(8,10,15,16)(18,25)(23,29)(24,32)(27,31)Polycyclic presentation:
2^9|a^2=hi,b^2=hi,c^2=hi,d^2=h,e^2=h,f^2=h,g^2=i,b^a=bf,c^a=cg,e^a=ei,f^a=fh,g^a=gi,d^b=df,e^b=eh,f^b=fh,g^c=gi,f^d=fh62 conjugacy classes — view the full character table.
| maximal subgroup of | #72283124, #72283723, #72283748, … |
| maximal normal subgroup of | #72283124, #72283723, #72283748, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6270271);