Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6366870 |
| 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 | |
1748 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)(3,8)(4,9)(5,7)(6,10)(11,16)(12,15)(13,14)(19,26)(20,21)(23,30)(24,28)(27,29)(31,32);(2,7)(3,11)(4,5)(8,15)(9,10)(12,13)(17,18)(19,23)(20,28)(21,24)(22,25)(26,30)(27,32)(29,31);(17,19,22,27)(18,23,25,32)(20,26,21,29)(24,31,28,30);(1,3,6,12)(2,8,10,15)(4,11,5,13)(7,14,9,16);(1,4,6,5)(2,7,10,9)(3,11,12,13)(8,14,15,16)(17,20,22,21)(18,24,25,28)(19,26,27,29)(23,31,32,30)Polycyclic presentation:
2^9|c^2=i,d^2=h,e^2=hi,f^2=h,g^2=i,b^a=bf,c^a=cg,e^a=ehi,f^a=fh,g^a=gi,d^b=df,e^b=ehi,f^b=fh,g^c=gi,f^d=fh62 conjugacy classes — view the full character table.
| maximal subgroup of | #72339730, #72339740, #72339747, … |
| maximal normal subgroup of | #72339730, #72339740, #72339747, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6274320);