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 | |
4932 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)(3,7)(4,5)(6,8)(17,18,22,23)(19,27,29,24)(20,30,31,25)(21,26,32,28);(1,2)(3,7)(4,5)(6,8)(9,10,13,14)(11,15,16,12)(18,23)(20,31)(24,27)(26,28);(2,5)(6,7)(10,14)(12,15)(18,23)(21,32)(24,27)(25,30);(3,8)(6,7)(10,14)(11,16)(17,19,22,29)(18,24,23,27)(20,26,31,28)(21,25,32,30);(1,3,4,8)(2,6,5,7)(9,11)(10,12)(13,16)(14,15)(17,20,22,31)(18,25,23,30)(19,28,29,26)(21,27,32,24);(1,2)(3,6)(4,5)(7,8)(9,12)(10,11)(13,15)(14,16)(17,21)(18,26)(19,25)(20,27)(22,32)(23,28)(24,31)(29,30)Polycyclic presentation:
2^9|a^2=g,b^2=ghi,d^2=g,e^2=h,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dghi,e^b=egi,f^c=fgi,e^d=egi92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8295445);