Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #493492 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #20585 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C2×#41 | C24 |
| prime radical | #62832 | C2 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C2×C4 ◁ #3498 ◁ G | |
| LECS | G ▷ C2×#41 ▷ C22 ▷ C2 ▷ C1 | |
2586 conjugacy classes of subgroups.
| minimal degree | μ(G) = 16 |
| transitive degree | μt(G) = 16 |
A minimal faithful permutation representation — 4 generators on 16 points:
(1,2)(3,10,12,6)(4,8)(5,9)(7,11,15,13)(14,16);(2,6)(7,8)(9,10)(15,16);(1,3)(2,7)(4,13)(5,12)(6,8)(9,15)(10,16)(11,14);(4,14)(7,15)(8,16)(11,13)Polycyclic presentation:
2^9|a^2=g,f^2=i,b^a=be,c^a=cf,e^a=eh,h^a=hi,g^b=gh,d^c=dh,f^c=fi,g^c=gi,f^d=fi,g^e=gi62 conjugacy classes — view the full character table.
| maximal subgroup of | #72225822, #72233665, #72234102, … |
| maximal normal subgroup of | #72225822, #72233665, #72234102, … |
| maximal quotient of | #72225792, #72226017, #72232751, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 400688);