Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| automorphism | ? |
| homology | Abelianization (H1) = C26 · Schur multiplier (H2) = C212 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C23 | C26 |
| prime radical | C2×H(3,𝔽4) | C22 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4695 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 32 points:
(1,2)(3,6)(4,5)(7,8)(9,10,12,14)(11,15,16,13)(17,18)(19,26)(20,21)(22,24)(23,25)(27,31)(28,29)(30,32);(9,11,12,16)(10,13,14,15)(17,18,23,25)(19,26,20,21)(22,24,30,32)(27,31,28,29);(2,5)(6,8)(17,19,23,20)(18,21,25,26)(22,28,30,27)(24,31,32,29);(1,3,4,7)(2,6,5,8)(17,20,23,19)(18,21,25,26)(22,27,30,28)(24,31,32,29);(1,3)(2,6)(4,7)(5,8)(17,21)(18,19)(20,25)(22,29)(23,26)(24,27)(28,32)(30,31);(1,2)(3,6)(4,5)(7,8)(17,22)(18,24)(19,27)(20,28)(21,29)(23,30)(25,32)(26,31)Polycyclic presentation:
2^9|a^2=gi,b^2=g,c^2=i,d^2=h,b^a=bg,c^a=ch,e^a=ei,c^b=ci,e^c=ei,f^c=fh,e^d=ehi,f^d=fi95 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7982861);