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 | |
5221 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)(17,18,22,20)(19,25,26,23)(21,28,29,24)(27,30,32,31);(1,2)(3,7)(4,5)(6,8)(10,11)(14,15)(18,20)(21,29)(23,25)(27,32);(2,5)(6,7)(9,11,13,10)(12,15,16,14)(18,20)(19,26)(24,28)(27,32);(9,12)(10,14)(11,15)(13,16)(17,19)(18,23)(20,25)(21,27)(22,26)(24,30)(28,31)(29,32);(1,2)(3,6)(4,5)(7,8)(17,20,22,18)(19,25,26,23)(21,28,29,24)(27,31,32,30);(1,3,4,8)(2,6,5,7)(17,21)(18,24)(19,27)(20,28)(22,29)(23,30)(25,31)(26,32)Polycyclic presentation:
2^9|a^2=g,c^2=gh,e^2=h,f^2=ghi,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,f^b=fgi,d^c=dh,e^c=egi80 conjugacy classes — view the full character table.
| maximal subgroup of | #72262811 |
| maximal normal subgroup of | #72262811 |
| maximal quotient of | #72262816, #72262829 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 9213183);