Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #45438 |
| 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 | C2×#127 | C2×C4 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1413 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)(3,7)(4,6)(5,8)(9,10)(11,16)(12,14)(13,15)(17,18,22,26,23,27,28,32)(19,24,20,25,29,21,30,31);(1,3,4,8)(2,5,6,7)(17,19)(18,24)(20,22)(21,27)(23,29)(25,26)(28,30)(31,32);(2,6)(5,7)(9,11,13,12)(10,14,15,16)(17,20)(18,25)(19,28)(21,26)(22,29)(23,30)(24,32)(27,31);(3,8)(5,7)(9,12,13,11)(10,14,15,16)(19,29)(20,30)(21,24)(25,31);(2,6)(5,7)(17,21,22,31,23,24,28,25)(18,20,26,29,27,30,32,19)Polycyclic presentation:
2^9|a^2=h,b^2=f,c^2=fg,d^2=fg,e^2=h,h^2=i,b^a=bf,c^a=cg,e^a=efi,c^b=ci,d^b=dfi,e^b=ei,d^c=di,e^d=ei104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6314027);