Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #40484 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C4 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, Frattini, cosocle | C22×C4 | C25 |
| commutator, socle | C23 | C24×C4 |
| prime radical | #2688 | C4 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1400 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)(3,6)(4,8)(5,7)(9,10)(11,14)(12,13)(15,16)(17,18,24,27,23,29,21,31)(19,30,20,22,28,25,32,26);(1,3,5,4)(2,6,7,8)(10,13)(11,15)(17,19)(18,25)(20,24)(21,32)(22,31)(23,28)(26,27)(29,30);(1,4,5,3)(2,6,7,8)(10,13)(14,16)(17,20)(18,26)(19,21)(22,29)(23,32)(24,28)(25,31)(27,30);(9,11,12,15)(10,14,13,16)(17,21,23,24)(18,27,29,31)(19,20,28,32)(22,30,26,25);(10,13)(14,16)(17,22,24,25,23,26,21,30)(18,28,27,32,29,19,31,20)Polycyclic presentation:
2^9|a^2=h,b^2=fg,c^2=fg,d^2=f,e^2=h,h^2=i,b^a=bf,c^a=cg,d^a=di,e^a=efi,c^b=ci,d^b=df,d^c=di,e^c=ei104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6312696);