Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6806903 |
| 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 | C22×#41 | C23 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C4×C22⋊Q8 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1359 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)(4,6)(5,8)(9,10,12,14)(11,15,16,13)(17,18)(19,29)(20,26)(21,28)(22,27)(23,31)(24,30)(25,32);(1,3)(2,5)(4,8)(6,7)(17,19)(18,23)(20,32)(21,30)(22,31)(24,25)(26,28)(27,29);(9,11,12,16)(10,13,14,15)(17,20,32,19,22,30,21,31)(18,24,25,23,27,26,28,29);(2,6)(5,7)(17,21,22,32)(18,25,27,28)(19,30,31,20)(23,24,29,26);(3,8)(5,7)(17,20,21,31,22,30,32,19)(18,26,28,23,27,24,25,29)Polycyclic presentation:
2^9|a^2=gi,c^2=gh,d^2=i,e^2=h,h^2=i,b^a=bf,c^a=cg,d^a=df,c^b=ch,e^b=efh,h^b=hi,d^c=di,e^c=ei,e^d=ei68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6725015);