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 | |
5129 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 32 points:
(1,2,5,3)(4,6,8,7)(10,12)(14,16);(2,3)(6,7)(11,15)(14,16)(17,18,22,23)(19,24,27,31)(20,28,30,21)(25,26,29,32);(4,8)(6,7)(9,10,13,12)(11,14,15,16)(18,23)(21,28)(24,31)(26,32);(2,3)(6,7)(9,11)(10,14)(12,16)(13,15)(17,19)(18,24)(20,29)(21,26)(22,27)(23,31)(25,30)(28,32);(1,3,5,2)(4,7,8,6)(9,12,13,10)(11,16,15,14)(17,20)(18,21)(19,25)(22,30)(23,28)(24,32)(26,31)(27,29);(1,4)(2,6)(3,7)(5,8)(17,21)(18,20)(19,26)(22,28)(23,30)(24,29)(25,31)(27,32)Polycyclic presentation:
2^9|a^2=g,b^2=i,c^2=h,e^2=gh,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^b=egi,f^c=fgi,e^d=egi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8258840);