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 | |
5068 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 32 points:
(9,10,13,15)(11,12,16,14)(17,18,23,25)(19,24,21,27)(20,26,31,30)(22,29,32,28);(1,2,5,4)(3,7,8,6)(9,11,13,16)(10,14,15,12)(17,19)(18,24)(20,29)(21,23)(22,30)(25,27)(26,32)(28,31);(1,3,5,8)(2,6,4,7)(9,11)(10,12)(13,16)(14,15)(18,25)(19,21)(20,31)(22,32);(3,8)(6,7)(9,12)(10,11)(13,14)(15,16)(17,20,23,31)(18,26,25,30)(19,29,21,28)(22,24,32,27);(1,3)(2,7)(4,6)(5,8)(17,21,23,19)(18,27,25,24)(20,29,31,28)(22,26,32,30);(1,4,5,2)(3,6,8,7)(17,22,23,32)(18,28,25,29)(19,30,21,26)(20,24,31,27)Polycyclic presentation:
2^9|a^2=gh,b^2=hi,c^2=ghi,d^2=h,e^2=h,f^2=gi,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, 8936229);