Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6437510 |
| 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×#532 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×C4×C4⋊C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1884 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 32 points:
(1,2,5,8)(3,7,4,6)(9,10)(11,14)(12,16)(13,15)(17,18,22,24)(19,28,20,25)(21,26,23,27)(29,31,30,32);(1,3,5,4)(2,6,8,7)(9,11,13,12)(10,14,15,16)(18,24)(19,29)(20,30)(21,23)(25,31)(28,32);(1,3,5,4)(2,7,8,6)(9,12,13,11)(10,14,15,16)(17,19,22,20)(18,25,24,28)(21,29,23,30)(26,32,27,31);(1,4,5,3)(2,6,8,7)(17,20,22,19)(18,25,24,28)(21,30,23,29)(26,32,27,31);(17,21,22,23)(18,26,24,27)(19,29,20,30)(25,32,28,31)Polycyclic presentation:
2^9|a^2=f,b^2=fg,c^2=fgi,d^2=f,e^2=i,h^2=i,b^a=bf,c^a=cg,c^b=ch,d^b=dh,e^b=ei,h^b=hi,h^c=hi,h^d=hi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6354494);