Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #4033837 |
| automorphism | Aut(G) order 1048576 |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C25×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C24 | C25 |
| prime radical | C25 | C24 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
1648 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)(3,7,8,4)(9,10,13,15)(11,16,14,12)(17,18,21,20)(19,22,23,24)(25,26,30,28)(27,31,29,32);(1,2)(3,7)(4,8)(5,6)(11,14)(12,16)(19,23)(22,24)(25,27,30,29)(26,31,28,32);(1,3,5,8)(2,4,6,7)(9,11,13,14)(10,12,15,16)(17,19,21,23)(18,22,20,24)(25,28,30,26)(27,31,29,32);(1,4,5,7)(2,3,6,8)(17,20,21,18)(19,24,23,22)(25,29,30,27)(26,31,28,32);(1,3)(2,4)(5,8)(6,7)(9,12)(10,14)(11,15)(13,16)Polycyclic presentation:
2^9|a^2=h,b^2=i,c^2=h,d^2=fgh,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,d^b=df,e^b=eg,e^c=ef,e^d=ef80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 3964481);