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 | |
5695 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)(9,10,13,15)(11,12,16,14);(1,2)(3,8)(4,6)(5,7)(9,11)(10,14)(12,15)(13,16)(17,18)(19,24)(20,28)(21,26)(22,23)(25,30)(27,31)(29,32);(1,3,4,7)(2,5,6,8)(10,15)(12,14)(18,23)(20,30)(24,32)(26,27);(2,6)(5,8)(9,11,13,16)(10,14,15,12)(17,19)(18,24)(20,27)(21,25)(22,29)(23,32)(26,30)(28,31);(1,3)(2,5)(4,7)(6,8)(9,12)(10,11)(13,14)(15,16)(17,20)(18,25)(19,27)(21,24)(22,30)(23,28)(26,29)(31,32);(1,2)(3,5)(4,6)(7,8)(9,11)(10,12)(13,16)(14,15)(17,21)(18,26)(19,28)(20,24)(22,31)(23,27)(25,29)(30,32)Polycyclic presentation:
2^9|a^2=g,c^2=gh,d^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^b=egi,e^c=egi,f^d=fgi89 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8490594);