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 | |
5103 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)(19,21)(20,28)(22,23)(24,26)(25,29)(27,32)(30,31);(1,2)(3,7)(4,5)(6,8)(10,11)(14,15)(18,23)(19,26)(20,29)(31,32);(2,5)(6,7)(9,11,13,10)(12,15,16,14)(18,23)(21,24)(25,28)(31,32);(3,8)(6,7)(9,12,13,16)(10,14,11,15)(17,19)(18,24)(20,30)(21,23)(22,26)(25,31)(27,29)(28,32);(1,3)(2,6)(4,8)(5,7)(17,20)(18,25)(19,27)(21,31)(22,29)(23,28)(24,32)(26,30);(1,2)(3,6)(4,5)(7,8)(17,21,22,24)(18,19,23,26)(20,31,29,32)(25,27,28,30)Polycyclic presentation:
2^9|a^2=gh,c^2=gh,d^2=gh,f^2=h,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^b=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, 8255500);