Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #394335 |
| 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 | |
837 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)(3,7)(4,8)(5,6)(9,10)(11,13)(12,14)(15,16)(17,18)(19,27)(20,23)(21,25)(22,26)(24,30)(28,31)(29,32);(2,6)(7,8)(9,11,12,15)(10,13,14,16)(18,23)(19,28)(20,21)(24,27)(25,26)(31,32);(1,3,5,4)(2,7,6,8)(17,19,20,29,22,28,21,30)(18,24,25,31,26,32,23,27);(1,4,5,3)(2,8,6,7)(10,14)(13,16)Polycyclic presentation:
2^9|b^2=h,c^2=e,d^2=ef,e^2=i,f^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,g^a=gi,c^b=cg,e^b=ei,f^b=fi,g^c=gi74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 314937);