Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6362022 |
| 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 | |
1571 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 32 points:
(1,2,5,7)(3,9,11,15)(4,13,14,8)(6,10,16,12)(17,18,22,24)(19,26,27,32)(20,25,21,29)(23,30,31,28);(1,3,5,11)(2,6,7,16)(4,10,14,12)(8,15,13,9)(19,27)(23,31)(26,32)(28,30);(1,2)(3,6)(4,8)(5,7)(9,12)(10,15)(11,16)(13,14)(17,19)(18,23)(20,28)(21,30)(22,27)(24,31)(25,32)(26,29);(3,10)(4,14)(6,15)(8,13)(9,16)(11,12)(18,24)(23,31)(25,29)(26,32);(2,7)(3,11)(8,13)(10,12)(17,20,22,21)(18,25,24,29)(19,28,27,30)(23,32,31,26)Polycyclic presentation:
2^9|a^2=hi,b^2=h,e^2=i,f^2=h,g^2=i,b^a=bf,c^a=cg,d^a=di,e^a=eh,f^a=fh,g^a=gi,c^b=ci,d^b=df,e^b=eh,f^b=fh,e^c=eh,g^c=gi,f^d=fh65 conjugacy classes — view the full character table.
| maximal subgroup of | #72339691, #72339721, #72339747, … |
| maximal normal subgroup of | #72339691, #72339721, #72339747, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6269293);