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 | |
6543 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 24 points:
(9,10)(11,15)(12,13)(14,16)(19,20)(22,24);(1,2)(3,8)(4,6)(5,7)(9,10,13,12)(11,14,16,15)(17,18,21,23)(19,24,20,22);(1,3)(2,4)(5,6)(7,8)(9,11)(10,14)(12,15)(13,16)(17,19,21,20)(18,22,23,24);(3,6)(4,8)(9,11,13,16)(10,14,12,15)(17,18,21,23)(19,24,20,22);(1,4,5,8)(2,6,7,3)(9,11)(10,15)(12,14)(13,16)(17,18)(19,22)(20,24)(21,23);(1,2)(3,4)(5,7)(6,8)(9,12,13,10)(11,15,16,14)(17,20,21,19)(18,22,23,24)Polycyclic presentation:
2^9|b^2=h,c^2=gh,d^2=h,e^2=ghi,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^d=egi,f^e=fgi89 conjugacy classes — view the full character table.
| maximal subgroup of | #72233534, #72233585, #72258018, … |
| maximal normal subgroup of | #72233534, #72233585, #72258018, … |
| maximal quotient of | #72260786, #72260845, #72263521, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 9068957);