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 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
5044 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 32 points:
(3,7)(5,8)(9,10)(11,12)(13,14)(15,16)(17,18,22,23)(19,27,28,24)(20,30,31,25)(21,26,32,29);(1,2,4,6)(3,5,7,8)(10,14)(11,15)(18,23)(19,28)(25,30)(26,29);(1,3)(2,5)(4,7)(6,8)(9,11,13,15)(10,12,14,16)(18,23)(20,31)(24,27)(26,29);(1,2)(3,5)(4,6)(7,8)(9,12)(10,15)(11,14)(13,16)(17,19)(18,24)(20,29)(21,25)(22,28)(23,27)(26,31)(30,32);(1,3)(2,5)(4,7)(6,8)(17,20)(18,25)(19,29)(21,24)(22,31)(23,30)(26,28)(27,32);(17,21)(18,26)(19,30)(20,24)(22,32)(23,29)(25,28)(27,31)Polycyclic presentation:
2^9|a^2=gi,b^2=ghi,c^2=i,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^c=egi,f^d=fgi83 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8438903);