Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6806527 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | C2×#466 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C4×C22⋊Q8 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1450 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 32 points:
(1,2,5,7)(3,6,4,8)(9,10,12,14)(11,15,16,13)(17,18,21,24)(19,26,20,23)(22,25,28,31)(27,32,29,30);(1,3,5,4)(2,6,7,8)(9,11)(10,13)(12,16)(14,15)(19,27)(20,29)(22,28)(23,30)(25,31)(26,32);(1,4,5,3)(2,6,7,8)(10,14)(11,16)(17,19,28,29,21,20,22,27)(18,23,31,32,24,26,25,30);(10,14)(13,15);(17,20,28,27,21,19,22,29)(18,23,31,32,24,26,25,30)Polycyclic presentation:
2^9|a^2=g,b^2=fgi,c^2=gh,e^2=fhi,h^2=i,b^a=bf,c^a=cg,d^a=df,c^b=ch,e^b=efh,h^b=hi86 conjugacy classes — view the full character table.
| maximal subgroup of | #72343783, #72343826, #72343833 |
| maximal normal subgroup of | #72343783, #72343826, #72343833 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6727956);