Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7514140 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C29 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #58990 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
3158 conjugacy classes of subgroups.
| minimal degree | μ(G) = 16 |
| transitive degree | μt(G) = 16 |
A minimal faithful permutation representation — 5 generators on 16 points:
(1,2)(3,11)(4,12)(5,8)(6,9)(7,10)(13,15)(14,16);(2,8)(3,12)(5,6)(9,10)(11,15)(13,16);(1,3)(2,4)(5,12)(6,16)(7,13)(8,15)(9,11)(10,14);(1,4)(2,3)(5,11)(6,15)(7,14)(8,16)(9,12)(10,13);(1,4)(2,3)(5,15)(6,11)(7,14)(8,12)(9,16)(10,13)Polycyclic presentation:
2^9|f^2=i,g^2=i,b^a=bf,c^a=cg,d^a=dg,e^a=ef,f^a=fi,g^a=gi,h^a=hi,c^b=cg,d^b=dh,f^b=fi,g^b=gi,g^c=gi,h^c=hi,g^d=gi,f^e=fi68 conjugacy classes — view the full character table.
| maximal subgroup of | #72209905, #72209906, #72213368, … |
| maximal normal subgroup of | #72209905, #72209906, #72213368, … |
| maximal quotient of | #72213250, #72215961, #72226293, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7421336);