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 | |
7138 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 6 generators on 24 points:
(3,7)(6,8)(10,14)(12,15)(17,18)(19,23)(20,21)(22,24);(1,2)(3,6)(4,5)(7,8)(9,10,13,14)(11,15,16,12);(2,5)(6,8)(10,14)(11,16)(18,21)(22,23);(17,19)(18,22)(20,24)(21,23);(1,2)(3,6)(4,5)(7,8)(9,11)(10,15)(12,14)(13,16)(17,18)(19,22)(20,21)(23,24);(1,3)(2,6)(4,7)(5,8)(9,12,13,15)(10,11,14,16)(17,19)(18,22)(20,24)(21,23)Polycyclic presentation:
2^9|b^2=g,f^2=g,b^a=bg,c^a=ch,d^a=dh,f^a=fhi,c^b=ci,e^c=ehi104 conjugacy classes — view the full character table.
| maximal subgroup of | #72232069, #72241222, #72262834, … |
| maximal normal subgroup of | #72232069, #72241222, #72262834, … |
| maximal quotient of | #72232079, #72232186, #72232539, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7549451);