Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #2805613 |
| 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, prime radical, cosocle | C24 | C25 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
1576 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 5 generators on 32 points:
(2,5)(3,8)(9,10)(11,15)(12,13)(14,16)(17,18,22,21)(19,20,24,23)(27,31)(29,32);(1,2,4,5)(3,6,8,7)(10,12)(11,16)(17,19,22,24)(18,23,21,20);(1,3)(2,6)(4,8)(5,7)(9,10,13,12)(11,14,16,15)(17,18,22,21)(19,23,24,20)(25,26,28,30)(27,29,31,32);(1,3)(2,7)(4,8)(5,6)(9,11)(10,14)(12,15)(13,16)(17,20)(18,19)(21,24)(22,23)(25,27,28,31)(26,29,30,32);(1,2)(3,6)(4,5)(7,8)(9,12,13,10)(11,14,16,15)(17,21,22,18)(19,23,24,20)Polycyclic presentation:
2^9|a^2=fg,b^2=fh,c^2=ghi,d^2=fi,e^2=fgh,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,d^b=dg,e^b=ef,e^d=ef80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 2868004);