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 | |
1971 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)(3,7)(4,6)(5,8)(10,15)(13,16)(17,18,20,22)(19,21,23,24)(25,26,30,29)(27,28,32,31);(1,2,4,6)(3,7,8,5)(9,10,14,15)(11,16,12,13)(19,23)(21,24)(25,27,30,32)(26,31,29,28);(3,8)(5,7)(9,11,14,12)(10,13,15,16)(17,18)(19,21)(20,22)(23,24)(25,27,30,32)(26,31,29,28);(1,2)(3,7)(4,6)(5,8)(9,12,14,11)(10,16,15,13)(17,19)(18,21)(20,23)(22,24)(25,28)(26,27)(29,32)(30,31);(1,3,4,8)(2,5,6,7)(9,13,14,16)(10,12,15,11)(17,18)(19,21)(20,22)(23,24)(25,29,30,26)(27,31,32,28)Polycyclic presentation:
2^9|a^2=g,b^2=f,c^2=hi,d^2=h,e^2=f,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, 3826843);