Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6667836 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | #36214 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C4×#121 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1916 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 32 points:
(1,2)(3,7)(4,6)(5,8)(9,10)(11,14)(12,16)(13,15)(17,18,20,23)(19,22,26,30)(21,24,28,32)(25,31,27,29);(1,3)(2,5)(4,8)(6,7)(9,11,13,12)(10,14,15,16)(17,19,20,26)(18,22,23,30)(21,25,28,27)(24,31,32,29);(9,12,13,11)(10,14,15,16)(18,23)(19,25)(21,28)(22,29)(26,27)(30,31);(2,6)(5,7)(19,26)(22,30)(25,27)(29,31);(3,8)(5,7)(19,25)(21,28)(22,31)(24,32)(26,27)(29,30)Polycyclic presentation:
2^9|a^2=i,b^2=g,c^2=gi,h^2=i,b^a=bf,c^a=cg,d^a=df,c^b=ch,d^b=di,e^b=efh,h^b=hi,h^c=hi,h^e=hi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6597387);