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 | |
5097 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)(19,25)(20,28)(21,23)(22,27)(24,30)(26,31)(29,32);(1,2,5,3)(4,6,8,7)(10,11)(14,15)(17,19,21,27)(18,22,23,25)(20,26,30,29)(24,31,28,32);(2,3)(6,7)(9,11,13,10)(12,15,16,14)(18,23)(22,25)(24,28)(31,32);(2,3)(4,8)(9,12,13,16)(10,14,11,15)(17,19)(18,22)(20,29)(21,27)(23,25)(24,32)(26,30)(28,31);(1,3,5,2)(4,7,8,6)(17,18)(19,25)(20,24)(21,23)(22,27)(26,32)(28,30)(29,31);(1,4)(2,6)(3,7)(5,8)(17,20)(18,24)(19,26)(21,30)(22,31)(23,28)(25,32)(27,29)Polycyclic presentation:
2^9|a^2=gh,b^2=gi,c^2=gh,d^2=gh,e^2=ghi,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,d^b=dgi,e^b=eh,e^c=egi,e^d=egi,f^d=fgi83 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 9012552);