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 | |
5087 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 32 points:
(9,10)(11,15)(12,14)(13,16)(17,18,22,23)(19,27,28,24)(20,25,30,29)(21,31,32,26);(1,2,5,4)(3,7,8,6)(9,10,12,14)(11,13,16,15)(18,23)(20,30)(24,27)(26,31);(1,3,5,8)(2,6,4,7)(9,10)(11,13)(12,14)(15,16)(18,23)(19,28)(20,30)(21,32);(3,8)(6,7)(9,11)(10,13)(12,16)(14,15)(17,19,22,28)(18,24,23,27)(20,31,30,26)(21,29,32,25);(1,3)(2,7)(4,6)(5,8)(17,20)(18,25)(19,26)(21,27)(22,30)(23,29)(24,32)(28,31);(1,4,5,2)(3,6,8,7)(17,21,22,32)(18,26,23,31)(19,29,28,25)(20,27,30,24)Polycyclic presentation:
2^9|a^2=h,b^2=i,c^2=ghi,d^2=h,f^2=gi,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dgi,e^c=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, 8935488);