Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6365694 |
| 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 | |
1697 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)(3,9)(4,13)(5,7)(6,12)(8,14)(10,16)(11,15)(17,18)(19,24)(20,29)(21,27)(22,31)(23,26)(25,30)(28,32);(1,3,5,11)(2,6,7,16)(4,10,14,12)(8,15,13,9)(17,19,23,20)(18,24,26,29)(21,30,22,28)(25,31,32,27);(17,20,23,19)(18,25,26,32)(21,30,22,28)(24,31,29,27);(3,10)(4,14)(6,15)(8,13)(9,16)(11,12)(18,26)(24,29)(25,32)(27,31);(2,7)(3,11)(8,13)(10,12)(17,21,23,22)(18,27,26,31)(19,28,20,30)(24,32,29,25)Polycyclic presentation:
2^9|b^2=hi,c^2=i,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,d^b=df,e^b=ehi,f^b=fh,g^c=gi,f^d=fh59 conjugacy classes — view the full character table.
| maximal subgroup of | #72339685, #72339696, #72339709, … |
| maximal normal subgroup of | #72339685, #72339696, #72339709, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6273340);