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 | |
4808 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 32 points:
(1,2,4,6)(3,5,7,8)(17,18,23,25)(19,27,29,24)(20,30,32,21)(22,31,28,26);(3,7)(5,8)(10,14)(11,16)(17,19,23,29)(18,24,25,27)(20,31,32,26)(21,28,30,22);(1,2)(3,5)(4,6)(7,8)(9,10)(11,15)(12,16)(13,14)(18,25)(21,30)(22,28)(24,27);(9,11,13,16)(10,15,14,12)(17,20,23,32)(18,21,25,30)(19,26,29,31)(22,27,28,24);(1,3)(2,5)(4,7)(6,8)(9,11)(10,12)(13,16)(14,15)(17,21)(18,20)(19,28)(22,29)(23,30)(24,31)(25,32)(26,27);(1,2)(3,5)(4,6)(7,8)(9,12)(10,11)(13,15)(14,16)(17,22,23,28)(18,26,25,31)(19,21,29,30)(20,27,32,24)Polycyclic presentation:
2^9|a^2=h,b^2=g,d^2=gi,f^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,d^b=dgi,e^b=eghi,e^c=egi,f^c=fgi,e^d=egi92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8948210);