Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7514196 |
| 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 | #2407 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C4×#123 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1662 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)(9,10,12,14)(11,13,15,16)(18,23)(24,31)(25,30)(26,32);(2,6)(7,8)(9,11)(10,13)(12,15)(14,16)(17,18,21,23)(19,26,28,32)(20,25,22,30)(24,27,31,29);(1,3,5,4)(2,7,6,8)(10,14)(11,15);(1,4,5,3)(2,8,6,7)(10,14)(13,16)(17,19,21,28)(18,24,23,31)(20,29,22,27)(25,32,30,26);(2,6)(7,8)(17,20,21,22)(18,25,23,30)(19,27,28,29)(24,26,31,32)Polycyclic presentation:
2^9|a^2=g,b^2=i,c^2=fi,d^2=f,e^2=i,h^2=i,b^a=bf,c^a=cg,d^a=dg,e^a=efi,c^b=cg,d^b=dh,h^b=hi,e^d=ei,h^d=hi77 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7433024);