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 | |
6272 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)(3,4)(5,7)(6,8)(18,21)(22,23);(1,3,5,8)(2,6,7,4)(9,10)(11,15)(12,14)(13,16)(17,18)(19,22)(20,21)(23,24);(9,11)(10,13)(12,16)(14,15)(17,18)(19,22)(20,21)(23,24);(2,7)(3,8)(10,14)(13,15)(17,18)(19,23)(20,21)(22,24);(1,4)(2,3)(5,6)(7,8)(9,11)(10,15)(12,16)(13,14)(17,18)(19,23)(20,21)(22,24);(1,2)(3,6)(4,8)(5,7)(9,10)(11,13)(12,14)(15,16)(17,19,20,24)(18,22,21,23)Polycyclic presentation:
2^9|b^2=gh,f^2=h,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dghi,f^d=fgi,f^e=fgi89 conjugacy classes — view the full character table.
| maximal subgroup of | #72232288, #72232449, #72232576, … |
| maximal normal subgroup of | #72232288, #72232449, #72232576, … |
| maximal quotient of | #72232373, #72232487, #72232624, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8597167);