Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #47826 |
| 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 | #2703 | C4 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1394 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)(3,5)(4,6)(7,8)(9,10,13,16)(11,15,12,14)(17,18,24,27,23,22,21,30)(19,26,28,31,20,25,29,32);(1,3,4,7)(2,5,6,8)(9,10)(11,14)(12,15)(13,16)(17,19)(18,25)(20,23)(21,29)(22,26)(24,28)(27,32)(30,31);(2,6)(5,8)(9,11,13,12)(10,14,16,15)(17,20,23,19)(18,26,22,25)(21,28,24,29)(27,31,30,32);(9,11,13,12)(10,15,16,14)(17,21,23,24)(18,27,22,30)(19,28,20,29)(25,31,26,32);(9,12,13,11)(10,15,16,14)(17,22,24,30,23,18,21,27)(19,25,28,32,20,26,29,31)Polycyclic presentation:
2^9|a^2=h,b^2=fg,c^2=f,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,e^b=ei,d^c=di,e^c=ei,e^d=ei104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6309329);