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 |
| prime radical | #34957 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4837 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 32 points:
(2,3)(6,7)(9,10)(11,15)(12,14)(13,16)(17,18,22,26)(19,23,27,32)(20,24,30,29)(21,25,31,28);(1,2,5,3)(4,6,8,7)(9,10,12,14)(11,13,16,15)(19,27)(20,30)(23,32)(24,29);(1,2,5,3)(4,7,8,6)(9,10)(11,13)(12,14)(15,16)(17,19)(18,23)(20,28)(21,24)(22,27)(25,30)(26,32)(29,31);(1,2,5,3)(4,6,8,7)(9,11)(10,13)(12,16)(14,15)(17,20)(18,24)(19,28)(21,23)(22,30)(25,27)(26,29)(31,32);(1,3,5,2)(4,7,8,6);(1,4)(2,6)(3,7)(5,8)(17,21)(18,25)(19,29)(20,23)(22,31)(24,27)(26,28)(30,32)Polycyclic presentation:
2^9|a^2=ghi,b^2=g,c^2=h,d^2=h,e^2=h,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dghi,f^c=fgi92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8086926);