Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #417647 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C22×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C2×#121 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C22×#41 | C23 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
807 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2,5,8)(3,6,4,7)(9,10,12,14)(11,13,15,16)(17,18,22,30,21,24,20,25)(19,26,27,23,29,32,28,31);(1,3,5,4)(2,6,8,7)(11,15)(13,16)(17,19,21,29)(18,23,24,31)(20,27,22,28)(25,32,30,26);(1,3,5,4)(2,7,8,6)(9,11,12,15)(10,13,14,16)(18,24)(23,31)(25,30)(26,32);(1,4,5,3)(2,7,8,6)(18,24)(19,27)(20,22)(23,26)(28,29)(31,32)Polycyclic presentation:
2^9|a^2=efg,b^2=f,c^2=fgi,d^2=fi,e^2=i,b^a=be,c^a=cf,d^a=dh,h^a=hi,c^b=cg,d^b=de,e^b=ei,h^b=hi,e^d=ei68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 353556);