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 | |
6235 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 6 generators on 24 points:
(1,2,6,4)(3,5,8,7)(11,15)(13,16)(19,24)(21,22);(1,3)(2,7)(4,5)(6,8)(9,10,12,14)(11,13,15,16)(17,18,20,23)(19,22,24,21);(9,11)(10,13)(12,15)(14,16)(17,19)(18,21)(20,24)(22,23);(2,4)(5,7)(17,18,20,23)(19,21,24,22);(1,4,6,2)(3,5,8,7)(9,10)(11,13)(12,14)(15,16)(17,18)(19,21)(20,23)(22,24);(1,5)(2,8)(3,4)(6,7)(9,11)(10,13)(12,15)(14,16)(17,19)(18,22)(20,24)(21,23)Polycyclic presentation:
2^9|a^2=g,b^2=h,d^2=i,e^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,e^d=egi,f^d=fgi,f^e=fgi89 conjugacy classes — view the full character table.
| maximal subgroup of | #72232290, #72280128, #72328066, … |
| maximal normal subgroup of | #72232290, #72280128, #72328066, … |
| maximal quotient of | #72232338, #72232351, #72304040, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 9130374);