Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #4417862 |
| automorphism | Aut(G) order 2097152 |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C29 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C24 | C25 |
| prime radical | C23×D4 | C23 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
3470 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 32 points:
(2,4)(6,8)(10,11)(14,15)(17,18,21,24)(19,22,25,29)(20,27,28,23)(26,32,31,30);(9,10,13,11)(12,14,16,15)(19,25)(22,29)(26,31)(30,32);(5,6)(7,8)(9,10,13,11)(12,14,16,15)(17,19)(18,22)(20,26)(21,25)(23,30)(24,29)(27,32)(28,31);(1,2)(3,4)(5,6)(7,8)(9,11,13,10)(12,14,16,15)(17,18)(19,22)(20,23)(21,24)(25,29)(26,30)(27,28)(31,32);(9,12,13,16)(10,14,11,15)(17,20,21,28)(18,23,24,27)(19,26,25,31)(22,30,29,32)Polycyclic presentation:
2^9|a^2=h,b^2=f,c^2=f,d^2=f,e^2=fh,b^a=bf,c^a=cg,d^a=di,e^a=eh,c^b=ch,e^d=ef80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 4327304);