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 | |
1634 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)(3,10,12,16)(4,9,6,13)(8,15,14,11)(17,18,21,24)(19,22,27,30)(20,23,28,25)(26,29,32,31);(1,3)(2,8)(4,11)(5,14)(6,15)(7,12)(9,16)(10,13)(17,19,21,27)(18,22,24,30)(20,26,28,32)(23,29,25,31);(1,4,7,6)(2,9,5,13)(3,11,12,15)(8,16,14,10)(18,23)(20,28)(22,29)(24,25)(26,32)(30,31);(1,5,7,2)(3,10,12,16)(4,13,6,9)(8,15,14,11);(3,12)(8,14)(10,16)(11,15)(18,24)(22,30)(23,25)(29,31)Polycyclic presentation:
2^9|a^2=hi,b^2=i,c^2=h,d^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 | #72306547, #72306715, #72306718, … |
| maximal normal subgroup of | #72306547, #72306715, #72306718, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6270213);