Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6996639 |
| 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 | #44869 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2388 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1742 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 32 points:
(1,2,5,4)(3,7,8,6)(9,10)(11,13)(12,14)(15,16)(17,18)(19,26)(20,31)(21,29)(22,28)(23,27)(24,25)(30,32);(1,3,5,8)(2,6,4,7)(9,10)(11,15)(12,14)(13,16)(17,19,22,25)(18,24,28,26)(20,29,32,27)(21,31,23,30);(1,4,5,2)(3,7,8,6)(9,11)(10,13)(12,16)(14,15)(17,20,23,24,22,32,21,26)(18,25,29,31,28,19,27,30);(3,8)(6,7)(11,16)(13,15)(17,19)(18,26)(20,29)(21,31)(22,25)(23,30)(24,28)(27,32);(10,14)(13,15)(17,21,22,23)(18,27,28,29)(19,30,25,31)(20,24,32,26)Polycyclic presentation:
2^9|a^2=fi,b^2=f,c^2=fgi,e^2=i,g^2=i,b^a=bf,c^a=cg,e^a=eh,g^a=gi,c^b=ch,d^b=df,e^b=eh,d^c=dh,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, 6927640);