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 | |
5971 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 24 points:
(9,10)(11,15)(12,14)(13,16)(17,18,20,21)(19,23,24,22);(1,2)(3,5)(4,6)(7,8)(9,11,14,16)(10,15,12,13)(18,21)(22,23);(2,3)(6,7)(10,12)(11,16)(17,18)(19,22)(20,21)(23,24);(4,8)(6,7)(11,16)(13,15)(17,18)(19,22)(20,21)(23,24);(1,3,5,2)(4,7,8,6)(9,12,14,10)(11,15,16,13)(17,19,20,24)(18,22,21,23);(1,4,5,8)(2,6,3,7)(9,13,14,15)(10,16,12,11)Polycyclic presentation:
2^9|a^2=g,b^2=gh,e^2=i,f^2=gi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^b=egi,e^c=egi,f^d=fgi89 conjugacy classes — view the full character table.
| maximal subgroup of | #72262649 |
| maximal normal subgroup of | #72262649 |
| maximal quotient of | #72262668 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8558232);