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 | |
4652 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)(19,24)(20,25)(21,26)(22,23)(27,31)(28,29)(30,32);(1,3)(2,6)(4,8)(5,7)(9,11,12,16)(10,13,14,15);(2,4)(3,7)(18,23)(21,31)(24,29)(25,32);(3,7)(6,8)(17,19)(18,24)(20,26)(21,25)(22,28)(23,29)(27,30)(31,32);(1,4,5,2)(3,8,7,6)(17,20)(18,25)(19,27)(21,29)(22,30)(23,32)(24,31)(26,28);(1,3)(2,6)(4,8)(5,7)(17,21,22,31)(18,26,23,27)(19,25,28,32)(20,29,30,24)Polycyclic presentation:
2^9|a^2=g,b^2=g,e^2=i,f^2=hi,b^a=bg,c^a=ch,e^a=ei,c^b=ci,d^b=di,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, 7998933);