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 | C2×#392 | C22 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
2176 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 32 points:
(1,2,5,4)(3,7,8,6)(9,10)(11,16)(12,13)(14,15)(17,18,20,22)(19,23,24,21)(25,26,29,30)(27,28,31,32);(1,3,5,8)(2,6,4,7)(10,15)(13,16)(19,24)(21,23)(25,27,29,31)(26,28,30,32);(9,11,14,12)(10,16,15,13)(17,18)(19,21)(20,22)(23,24)(25,28)(26,27)(29,32)(30,31);(1,4,5,2)(3,7,8,6)(9,12,14,11)(10,16,15,13)(17,19)(18,21)(20,24)(22,23)(25,27)(26,28)(29,31)(30,32);(1,3)(2,7)(4,6)(5,8)(9,13,14,16)(10,12,15,11)(17,18)(19,21)(20,22)(23,24)(25,26)(27,28)(29,30)(31,32)Polycyclic presentation:
2^9|a^2=fhi,b^2=fgi,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.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 2974496);