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, cosocle | C24 | C25 |
| prime radical | #15263 | C2 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
2589 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)(3,5)(4,7)(6,8)(9,10,14,13)(11,16,12,15)(18,21)(19,23)(27,28)(30,31);(2,3)(4,8)(9,11,14,12)(10,15,13,16)(19,23)(22,24)(25,26,29,32)(27,30,28,31);(1,2,5,3)(4,6,8,7)(9,11,14,12)(10,16,13,15)(17,18)(19,22)(20,21)(23,24)(25,27,29,28)(26,30,32,31);(1,3,5,2)(4,6,8,7)(9,12,14,11)(10,16,13,15)(17,19)(18,22)(20,23)(21,24)(25,26)(27,30)(28,31)(29,32);(1,4,5,8)(2,6,3,7)(9,13,14,10)(11,15,12,16)(17,18)(19,22)(20,21)(23,24)(25,28,29,27)(26,31,32,30)Polycyclic presentation:
2^9|a^2=fh,b^2=gi,c^2=ghi,d^2=f,e^2=ghi,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.
| maximal subgroup of | #72303423, #72303506, #72303510, … |
| maximal normal subgroup of | #72303423, #72303506, #72303510, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 2958023);