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) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C24 | C25 |
| prime radical | #15273 | C2 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
2460 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 32 points:
(2,6)(3,7)(9,10)(11,12)(13,15)(14,16)(18,20)(19,24)(25,26,29,30)(27,31,32,28);(1,2,5,6)(3,8,7,4)(9,11,14,15)(10,13,16,12)(17,18)(19,22)(20,21)(23,24)(25,26)(27,31)(28,32)(29,30);(2,6)(4,8)(9,12,14,13)(10,11,16,15)(17,19,21,24)(18,22,20,23)(25,27,29,32)(26,28,30,31);(1,3,5,7)(2,4,6,8)(9,13,14,12)(10,15,16,11)(17,20,21,18)(19,23,24,22)(25,28,29,31)(26,27,30,32);(1,4)(2,3)(5,8)(6,7)(9,10)(11,13)(12,15)(14,16)(17,19)(18,23)(20,22)(21,24)(25,27)(26,28)(29,32)(30,31)Polycyclic presentation:
2^9|a^2=fgh,b^2=fg,c^2=hi,d^2=f,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, 3971172);