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 | #2733 | 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)(11,14)(12,16)(13,15)(17,18,24,26,23,28,21,30)(19,22,27,32,20,25,29,31);(1,3,4,7)(2,5,6,8)(9,11,14,10)(12,16,13,15)(17,19,23,20)(18,25,28,22)(21,29,24,27)(26,31,30,32);(2,6)(5,8)(9,12,14,13)(10,15,11,16)(17,20,23,19)(18,22,28,25)(21,27,24,29)(26,32,30,31);(9,12,14,13)(10,16,11,15)(17,21,23,24)(18,26,28,30)(19,27,20,29)(22,31,25,32);(9,13,14,12)(10,16,11,15)(17,22,21,31,23,25,24,32)(18,27,30,19,28,29,26,20)Polycyclic presentation:
2^9|a^2=h,b^2=g,c^2=f,d^2=f,e^2=fh,h^2=i,b^a=bf,c^a=cg,d^a=di,e^a=efi,d^b=df,e^b=ei,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, 6310256);