Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #5035983 |
| 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 | #15267 | C2 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
2493 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 32 points:
(1,2)(3,5)(4,6)(7,8)(9,10)(11,15)(12,16)(13,14)(17,18,21,20)(19,23,24,22)(25,26)(27,31)(28,29)(30,32);(4,8)(6,7)(10,14)(11,16)(17,19,21,24)(18,22,20,23)(25,27,30,28)(26,31,32,29);(2,3)(6,7)(11,16)(12,15)(18,20)(22,23)(26,32)(27,28);(1,3,5,2)(4,7,8,6)(9,11,13,16)(10,15,14,12)(17,19)(18,23)(20,22)(21,24)(25,28,30,27)(26,31,32,29);(1,4)(2,6)(3,7)(5,8)(9,12,13,15)(10,16,14,11)(17,20,21,18)(19,22,24,23)(25,29,30,31)(26,27,32,28)Polycyclic presentation:
2^9|a^2=gi,b^2=ghi,d^2=fg,e^2=fh,b^a=bf,c^a=cg,d^a=di,e^a=eh,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, 4981379);