Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6667700 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C26×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | C22×#121 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×C4×C22⋊C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1718 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)(17,18,21,26,22,24,20,25)(19,27,28,31,29,32,30,23);(1,3,4,8)(2,5,6,7)(9,10)(11,15)(12,14)(13,16);(3,8)(5,7)(9,11,12,16)(10,13,14,15)(17,19)(18,23)(20,28)(21,30)(22,29)(24,31)(25,27)(26,32);(2,6)(5,7)(18,24)(23,31)(25,26)(27,32);(10,14)(13,15)(17,20,22,21)(18,25,24,26)(19,28,29,30)(23,27,31,32)Polycyclic presentation:
2^9|a^2=g,b^2=f,c^2=fh,e^2=i,g^2=i,b^a=bf,c^a=cg,d^a=dfi,c^b=ch,e^b=efh,g^c=gi71 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6665169);