Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6573615 |
| 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×#449 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2296 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1520 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)(3,6)(4,7)(5,8)(17,18,22,28,23,27,21,26)(19,29,30,32,20,25,31,24);(1,3,5,4)(2,6,8,7)(9,10,12,13)(11,14,15,16)(17,19,23,20)(18,24,27,32)(21,30,22,31)(25,26,29,28);(1,3,5,4)(2,7,8,6)(10,13)(11,15)(17,20,23,19)(18,25,27,29)(21,30,22,31)(24,26,32,28);(1,3,5,4)(2,6,8,7)(9,11)(10,14)(12,15)(13,16)(17,21,23,22)(18,26,27,28)(19,30,20,31)(24,29,32,25);(1,4,5,3)(2,7,8,6)(10,13)(11,15)(18,27)(24,32)(25,29)(26,28)Polycyclic presentation:
2^9|a^2=f,b^2=ghi,c^2=gi,d^2=gi,e^2=g,f^2=i,b^a=bf,c^a=cg,e^a=ei,c^b=ch,e^b=eh,f^b=fi,d^c=dhi,e^d=eh77 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6524342);