Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #3858029 |
| automorphism | Aut(G) order 1048576 |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C24 | C25 |
| prime radical | #14885 | C2 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
2870 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 32 points:
(1,2,5,3)(4,7,8,6)(9,10)(11,15)(12,13)(14,16)(18,21)(19,23)(26,30)(28,31);(2,3)(4,8)(10,13)(14,15)(17,18,20,21)(19,24,23,22)(25,26)(27,31)(28,32)(29,30);(4,8)(6,7)(10,13)(14,15)(18,21)(22,24)(26,30)(28,31);(1,3,5,2)(4,6,8,7)(9,11,12,16)(10,14,13,15)(17,19,20,23)(18,22,21,24)(25,27,29,32)(26,28,30,31);(1,4)(2,6)(3,7)(5,8)(9,10)(11,15)(12,13)(14,16)(17,18)(19,24)(20,21)(22,23)(25,28)(26,27)(29,31)(30,32)Polycyclic presentation:
2^9|a^2=fgh,b^2=gi,d^2=f,b^a=bf,c^a=cg,d^a=di,e^a=efg,c^b=ch,d^b=dfg,e^b=ef,e^c=ef,e^d=ef86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 3782866);