Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of D8×#120 |
| 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×#548 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C4×#124 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1582 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)(11,13)(12,14)(15,16);(1,3,5,4)(2,6,8,7)(9,11,12,15)(10,13,14,16)(18,23)(20,29)(21,31)(24,25)(26,30)(27,32);(1,3,5,4)(2,7,8,6)(10,14)(13,16)(17,18,22,20)(19,25,28,26)(21,29,31,23)(24,27,30,32);(1,4,5,3)(2,6,8,7)(17,19,22,28)(18,24,20,30)(21,32,31,27)(23,25,29,26);(17,20,22,18)(19,26,28,25)(21,23,31,29)(24,32,30,27)Polycyclic presentation:
2^9|b^2=fg,c^2=fi,d^2=fi,e^2=i,h^2=i,b^a=bf,c^a=cg,c^b=ch,e^b=eh,h^b=hi,d^c=dhi,h^c=hi,e^d=eh,h^d=hi,h^e=hi98 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6488513);