Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #62295 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C4 · Schur multiplier (H2) = C29 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C23×D4 |
| commutator | C2×C4 | C24×C4 |
| Frattini, cosocle | C42 | C25 |
| socle | C22 | C23×C22⋊C4 |
| prime radical | #62781 | C2 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C2×C4×#41 ◁ G | |
| LECS | G ▷ C42 ▷ C22 ▷ C1 | |
2883 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 32 points:
(2,7)(4,5)(8,14)(9,13)(11,12)(15,16)(17,18,21,25,22,26,20,24)(19,27,28,23,29,31,30,32);(1,2)(3,8)(4,7)(5,13)(6,9)(10,15)(11,14)(12,16)(17,19)(18,23)(20,30)(21,28)(22,29)(24,27)(25,31)(26,32);(1,3)(2,8)(4,11)(5,12)(6,10)(7,14)(9,15)(13,16);(3,10)(8,15)(11,12)(14,16);(1,4,6,5)(2,7,9,13)(3,11,10,12)(8,14,15,16)(17,20,22,21)(18,24,26,25)(19,28,29,30)(23,31,32,27)Polycyclic presentation:
2^9|a^2=g,e^2=h,f^2=h,g^2=i,b^a=bf,e^a=ehi,f^a=fh,e^b=ei,f^b=fh,d^c=dhi98 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6250452);