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, prime radical, cosocle | C24 | C25 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
1498 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 5 generators on 32 points:
(1,2,4,7)(3,6,8,5)(9,10)(11,14)(12,16)(13,15)(19,23)(22,24)(25,26,29,27)(28,31,32,30);(1,3,4,8)(2,5,7,6)(9,10,13,15)(11,12,16,14)(18,21)(22,24);(1,3,4,8)(2,5,7,6)(9,11,13,16)(10,14,15,12)(17,18,20,21)(19,22,23,24)(25,27,29,26)(28,31,32,30);(1,2)(3,5)(4,7)(6,8)(9,10)(11,14)(12,16)(13,15)(17,19,20,23)(18,22,21,24)(25,28,29,32)(26,30,27,31);(1,3)(2,6)(4,8)(5,7)(9,12,13,14)(10,11,15,16)Polycyclic presentation:
2^9|a^2=fghi,b^2=f,c^2=fgi,d^2=gi,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, 3919968);