Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #40484 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C4 · Schur multiplier (H2) = C26×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, Frattini, cosocle | C22×C4 | C25 |
| commutator, socle | C23 | C24×C4 |
| prime radical | C23×D4 | C2×C4 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1878 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)(12,13)(15,16)(17,18,22,23,21,25,19,26)(20,28,30,32,29,24,27,31);(1,3,4,8)(2,5,6,7)(17,19,21,22)(18,23,25,26)(20,27,29,30)(24,31,28,32);(3,8)(5,7)(10,13)(14,16);(2,6)(5,7)(9,11,12,15)(10,14,13,16)(17,20)(18,24)(19,27)(21,29)(22,30)(23,31)(25,28)(26,32);(20,29)(24,28)(27,30)(31,32)Polycyclic presentation:
2^9|a^2=h,b^2=f,d^2=g,h^2=i,b^a=bf,c^a=cg,d^a=df,c^b=cfi,e^d=ei104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6282360);