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 | |
4873 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)(3,8)(4,7)(5,6)(9,10)(11,15)(12,14)(13,16)(18,23)(19,26)(25,30)(27,32);(3,4)(7,8)(9,10,12,14)(11,13,16,15)(17,18,22,23)(19,24,26,20)(21,25,29,30)(27,31,32,28);(1,3)(2,7)(4,6)(5,8)(9,10)(11,13)(12,14)(15,16)(19,26)(20,24)(21,29)(25,30);(3,4)(7,8)(9,11)(10,13)(12,16)(14,15)(17,19,22,26)(18,24,23,20)(21,27,29,32)(25,31,30,28);(1,4,6,3)(2,7,5,8)(17,20,22,24)(18,19,23,26)(21,28,29,31)(25,27,30,32);(1,5,6,2)(3,7,4,8)(17,21)(18,25)(19,27)(20,28)(22,29)(23,30)(24,31)(26,32)Polycyclic presentation:
2^9|b^2=g,d^2=ghi,e^2=gi,f^2=h,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dgi,e^c=egi,f^c=fgi,e^d=egi86 conjugacy classes — view the full character table.
| maximal subgroup of | #72303333 |
| maximal normal subgroup of | #72303333 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8916474);