Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #5211343 |
| automorphism | Aut(G) order 1048576 |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C24 | C25 |
| prime radical | #15263 | C2 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
2101 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 32 points:
(1,2,4,5)(3,7,8,6)(10,13)(14,16)(17,18)(19,23)(20,22)(21,24)(25,26,28,30)(27,29,31,32);(2,5)(3,8)(9,10,12,13)(11,14,15,16)(17,19)(18,23)(20,21)(22,24)(27,31)(29,32);(2,5)(6,7)(10,13)(11,15)(18,20)(19,24)(25,27)(26,29)(28,31)(30,32);(1,3,4,8)(2,6,5,7)(9,10)(11,14)(12,13)(15,16)(17,20,22,18)(19,21,24,23);(1,2)(3,6)(4,5)(7,8)(9,11)(10,14)(12,15)(13,16)(17,21)(18,19)(20,24)(22,23)(25,26)(27,29)(28,30)(31,32)Polycyclic presentation:
2^9|a^2=hi,b^2=gi,d^2=g,b^a=bf,c^a=cg,d^a=di,e^a=efh,c^b=ch,d^b=df,e^b=eg,d^c=dfg,e^c=ef68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 5143687);