Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #2760365 |
| automorphism | Aut(G) order 1048576 |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C24×C42 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C24 | C25 |
| prime radical | #15267 | C2 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
2764 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 32 points:
(2,5)(3,7)(11,15)(14,16)(17,18,22,20)(19,24,21,23)(25,26,30,32)(27,28,29,31);(1,2,4,5)(3,6,7,8)(9,10,13,12)(11,14,15,16)(17,19,22,21)(18,23,20,24)(25,27,30,29)(26,31,32,28);(2,5)(6,8)(18,20)(23,24)(25,28,30,31)(26,27,32,29);(1,2)(3,8)(4,5)(6,7)(9,11)(10,14)(12,16)(13,15)(17,20,22,18)(19,24,21,23)(25,28)(26,27)(29,32)(30,31);(1,3,4,7)(2,6,5,8)(9,12,13,10)(11,16,15,14)(17,21,22,19)(18,23,20,24)(25,29,30,27)(26,31,32,28)Polycyclic presentation:
2^9|a^2=g,b^2=fhi,c^2=fgh,d^2=fh,e^2=fhi,b^a=bf,c^a=cg,d^a=di,e^a=efg,c^b=ch,d^b=dfg,d^c=df,e^d=ef74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 2711879);