Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6533195 |
| 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×#540 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2298 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1468 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)(3,7)(4,8)(5,6)(17,18,22,26)(19,27,29,31)(20,25,21,23)(24,30,32,28);(2,6)(7,8)(9,10)(11,14)(12,13)(15,16)(18,23)(20,21)(24,27)(25,26)(28,30)(31,32);(10,13)(14,16)(17,19)(18,24)(20,28)(21,30)(22,29)(23,27)(25,31)(26,32);(1,3,5,4)(2,7,6,8)(9,11)(10,14)(12,15)(13,16)(17,20,22,21)(18,25,26,23)(19,28,29,30)(24,31,32,27);(1,4,5,3)(2,8,6,7)(11,15)(14,16)Polycyclic presentation:
2^9|a^2=i,d^2=fg,e^2=fgi,f^2=i,g^2=i,b^a=bf,c^a=cg,f^a=fi,g^a=gi,c^b=ch,d^b=di,f^b=fi,g^b=gi,f^c=fi,g^c=gi,e^d=eh77 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6442024);