Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #3858029 |
| 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 | C23×D4 | C23 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
2018 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)(3,4,7,8)(10,11)(12,16)(17,18,20,23)(19,22,24,21)(26,28)(27,31);(2,6)(4,8)(10,11)(14,15)(17,19,20,24)(18,21,23,22)(27,31)(30,32);(2,6)(4,8)(9,10,13,11)(12,14,16,15)(17,18)(19,21)(20,23)(22,24)(25,26,29,28)(27,30,31,32);(1,3)(2,4)(5,7)(6,8)(9,11,13,10)(12,15,16,14)(17,18)(19,21)(20,23)(22,24)(25,27)(26,30)(28,32)(29,31);(1,4)(2,7)(3,6)(5,8)(9,12,13,16)(10,14,11,15)(17,19)(18,22)(20,24)(21,23)(25,28,29,26)(27,32,31,30)Polycyclic presentation:
2^9|a^2=f,b^2=fgi,c^2=fg,d^2=h,e^2=fg,b^a=bf,c^a=cg,d^a=di,e^a=efg,c^b=ch,d^b=dfg,e^b=ef,e^c=ef,e^d=ef86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 3790656);