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 | |
5809 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)(3,7)(4,6)(5,8)(10,12)(14,16)(17,18)(19,23)(20,22)(21,24);(1,3)(2,5)(4,8)(6,7)(11,15)(14,16)(17,18,20,22)(19,21,24,23);(9,10,13,12)(11,14,15,16)(17,18)(19,21)(20,22)(23,24);(1,2,4,6)(3,5,8,7)(9,11,13,15)(10,14,12,16)(17,19,20,24)(18,21,22,23);(1,3)(2,5)(4,8)(6,7)(9,12,13,10)(11,16,15,14);(1,2)(3,5)(4,6)(7,8)Polycyclic presentation:
2^9|b^2=i,c^2=h,d^2=gh,e^2=h,b^a=bg,c^a=ch,d^a=dg,e^a=eghi,c^b=ci,d^b=dh,f^d=fgi89 conjugacy classes — view the full character table.
| maximal subgroup of | #72330757 |
| maximal normal subgroup of | #72330757 |
| maximal quotient of | #72330822, #72331799, #72331971, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8342291);