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 | |
4954 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 32 points:
(9,10,13,14)(11,12,15,16)(17,18,22,27)(19,25,28,23)(20,24,26,21)(29,30,31,32);(1,2)(3,4)(5,6)(7,8)(10,14)(11,15)(19,28)(23,25)(29,31)(30,32);(2,6)(3,7)(9,11)(10,12)(13,15)(14,16)(17,19,22,28)(18,23,27,25)(20,29,26,31)(21,30,24,32);(3,7)(4,8)(9,12)(10,11)(13,16)(14,15)(17,20)(18,24)(19,29)(21,27)(22,26)(23,32)(25,30)(28,31);(1,3)(2,4)(5,7)(6,8)(17,19)(18,25)(20,31)(21,30)(22,28)(23,27)(24,32)(26,29);(1,4)(2,3)(5,8)(6,7)(17,21,22,24)(18,26,27,20)(19,30,28,32)(23,31,25,29)Polycyclic presentation:
2^9|a^2=gh,c^2=h,f^2=h,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,e^c=egi,e^d=egi,f^d=fgi89 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8972655);