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 | |
5113 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 32 points:
(9,10,13,11)(12,15,16,14)(17,18)(19,27)(20,23)(21,25)(22,30)(24,28)(26,32)(29,31);(1,2)(3,8)(4,6)(5,7)(10,11)(14,15)(17,19,23,28)(18,24,20,27)(21,30,31,26)(22,25,32,29);(1,3)(2,4)(5,6)(7,8)(9,11,13,10)(12,15,16,14)(18,20)(22,32)(24,27)(25,29);(3,6)(4,8)(9,12)(10,14)(11,15)(13,16)(17,20,23,18)(19,27,28,24)(21,25,31,29)(22,30,32,26);(1,4,5,8)(2,6,7,3)(17,21,23,31)(18,25,20,29)(19,26,28,30)(22,27,32,24);(1,2)(3,8)(4,6)(5,7)(17,22,23,32)(18,26,20,30)(19,25,28,29)(21,27,31,24)Polycyclic presentation:
2^9|a^2=gh,b^2=h,c^2=gh,d^2=h,e^2=gi,f^2=h,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dgi,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, 8821416);