Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7500661 |
| 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 | C22×D4 | C24 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C42×Q8 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1339 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)(3,4)(5,7)(6,8)(9,10,13,16)(11,15,12,14)(17,18)(19,22)(20,24)(21,23)(25,29)(26,30)(27,31)(28,32);(1,3)(2,6)(4,7)(5,8)(17,19,21,26)(18,22,23,30)(20,25,28,27)(24,29,32,31);(9,11,13,12)(10,14,16,15)(19,25)(20,28)(22,29)(24,32)(26,27)(30,31);(2,7)(3,8)(9,12,13,11)(10,15,16,14)(18,23)(22,30)(24,32)(29,31);(1,4,5,6)(2,3,7,8)(19,26)(22,30)(25,27)(29,31)Polycyclic presentation:
2^9|a^2=g,b^2=i,c^2=g,d^2=g,e^2=f,h^2=i,b^a=bf,c^a=cg,d^a=dfgi,c^b=ch,d^b=df,e^b=ei,h^b=hi,h^c=hi74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7418916);