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 | |
4688 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)(3,5)(4,7)(6,8)(9,10,13,15)(11,12,16,14)(17,18,23,25)(19,24,29,32)(20,26,30,31)(21,27,28,22);(1,2,6,8)(3,5,7,4)(9,11,13,16)(10,14,15,12)(17,19,23,29)(18,24,25,32)(20,28,30,21)(22,31,27,26);(1,3,6,7)(2,4,8,5)(9,11)(10,12)(13,16)(14,15)(18,25)(19,29)(20,30)(22,27);(1,4,6,5)(2,3,8,7)(9,12)(10,11)(13,14)(15,16)(17,20)(18,26)(19,28)(21,29)(22,24)(23,30)(25,31)(27,32);(1,2)(3,4)(5,7)(6,8)(17,21,23,28)(18,27,25,22)(19,30,29,20)(24,31,32,26);(1,5,6,4)(2,7,8,3)(17,22,23,27)(18,28,25,21)(19,31,29,26)(20,24,30,32)Polycyclic presentation:
2^9|a^2=gh,b^2=i,c^2=ghi,d^2=ghi,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, 8831859);