Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #235462 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×#392 |
| commutator, Frattini, cosocle | C2×C42 | C24 |
| prime radical | #35237 | C2 |
| LCS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
1526 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2,6,10)(3,11,12,15)(4,7,5,9)(8,14,16,13)(17,18,21,24)(19,25,27,32)(20,28,30,23)(22,29,31,26);(2,7)(4,5)(8,11)(9,10)(13,14)(15,16)(17,19)(18,22)(20,29)(21,27)(23,32)(24,31)(25,28)(26,30);(1,3)(2,8)(4,13)(5,14)(6,12)(7,15)(9,11)(10,16)(20,30)(23,28)(25,32)(26,29);(3,12)(8,16)(11,15)(13,14)Polycyclic presentation:
2^9|a^2=h,e^2=h,f^2=hi,b^a=be,c^a=cf,e^a=eh,f^a=fhi,g^a=ghi,c^b=cg,e^b=eh,f^b=fh,d^c=dhi,e^c=ei,f^c=fhi62 conjugacy classes — view the full character table.
| maximal subgroup of | #72306361, #72306365, #72306370, … |
| maximal normal subgroup of | #72306361, #72306365, #72306370, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 142659);