Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #20158 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | D42 |
| commutator | C42 | C23×C4 |
| Frattini, cosocle | C2×C42 | C24 |
| socle | C22 | D4×C4⋊C4 |
| prime radical | C2×#135 | Q8 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
513 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2,4,10,7,3,6,8)(5,9,12,13,14,11,15,16)(17,18,24,27)(19,28,30,21)(20,22,26,23)(25,31,32,29);(1,3,7,2)(4,8,6,10)(5,13,14,16)(9,12,11,15)(17,19,24,30)(18,25,27,32)(20,21,26,28)(22,31,23,29);(1,4,7,6)(2,8,3,10)(5,12,14,15)(9,16,11,13)(17,20,23,18,24,26,22,27)(19,21,29,25,30,28,31,32);(1,5,7,14)(2,9,3,11)(4,12,6,15)(8,16,10,13)(17,21,24,28)(18,19,27,30)(20,31,26,29)(22,32,23,25)Polycyclic presentation:
2^9|a^2=g,b^2=h,c^2=f,d^2=h,e^2=h,f^2=i,g^2=hi,b^a=be,c^a=cf,e^a=eh,f^a=fi,c^b=chi,d^b=de,e^b=eh,d^c=di,e^d=eh68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 132991);