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 | #15630 | C2 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
2190 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 32 points:
(1,2,5,6)(3,4,8,7)(9,10,13,12)(11,14,16,15)(17,18)(19,21)(20,22)(23,24)(27,31)(29,32);(1,3)(2,4)(5,8)(6,7)(9,10)(11,15)(12,13)(14,16)(18,19)(20,24)(25,26,28,30)(27,29,31,32);(2,6)(3,8)(10,12)(14,15)(17,18,21,19)(20,23,24,22)(25,27,28,31)(26,29,30,32);(1,3)(2,7)(4,6)(5,8)(9,11,13,16)(10,14,12,15)(17,19,21,18)(20,23,24,22)(25,27)(26,29)(28,31)(30,32);(1,4,5,7)(2,3,6,8)(9,12,13,10)(11,15,16,14)(17,20)(18,22)(19,23)(21,24)(25,26)(27,29)(28,30)(31,32)Polycyclic presentation:
2^9|a^2=fghi,b^2=gh,c^2=fhi,d^2=f,e^2=fghi,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, 5372376);