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 | |
7755 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)(10,14)(11,16)(18,21)(19,23);(1,3)(2,5)(4,8)(6,7)(9,10)(11,15)(12,16)(13,14)(17,18,20,21)(19,22,23,24);(9,11)(10,15)(12,14)(13,16)(17,18)(19,22)(20,21)(23,24);(2,6)(5,7)(9,11)(10,12)(13,16)(14,15)(17,18)(19,22)(20,21)(23,24);(1,3)(2,5)(4,8)(6,7)(17,19)(18,22)(20,23)(21,24);(1,2)(3,5)(4,6)(7,8)(9,12)(10,16)(11,14)(13,15)Polycyclic presentation:
2^9|b^2=i,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 | #72232068, #72232278, #72232845, … |
| maximal normal subgroup of | #72232068, #72232278, #72232845, … |
| maximal quotient of | #72232134, #72232340, #72232368, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8340869);