Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #4307937 |
| automorphism | Aut(G) order 402653184 |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, prime radical, cosocle | C24 | C25 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
2754 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 5 generators on 32 points:
(2,4)(6,7)(9,10,12,14)(11,15,16,13)(20,27)(24,29)(25,31)(30,32);(5,6,8,7)(18,22)(23,28)(24,29)(30,32);(1,2,3,4)(5,7,8,6)(17,18)(19,23)(20,24)(21,22)(25,30)(26,28)(27,29)(31,32);(9,11,12,16)(10,13,14,15)(17,19,21,26)(18,23,22,28)(20,25,27,31)(24,30,29,32);(17,20,21,27)(18,24,22,29)(19,25,26,31)(23,30,28,32)Polycyclic presentation:
2^9|a^2=i,b^2=f,c^2=g,d^2=hi,e^2=h,b^a=bf,c^a=cg,d^a=di,e^a=eh,c^b=ch104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 4215710);