Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #511835 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #32898 |
| commutator, Frattini, cosocle | C2×#41 | C24 |
| LCS | G ▷ C2×#41 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C2×C4 ◁ C2×H(5,𝔽2) ◁ G | |
| LECS | G ▷ C2×#41 ▷ C22 ▷ C2 ▷ C1 | |
2685 conjugacy classes of subgroups; only the 186 normal subgroups are shown.
This group has 2685 conjugacy classes of subgroups — too many to draw. Only its 186 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 16 |
| transitive degree | μt(G) = 16 |
A minimal faithful permutation representation — 4 generators on 16 points:
(1,2)(3,12)(4,15)(5,10)(6,16)(7,11)(8,13)(9,14);(1,3,8,7)(2,9,13,15)(4,5,14,6)(10,11,16,12);(1,4,8,14)(2,10,13,16)(3,6,7,5)(9,12,15,11);(1,5,8,6)(2,9,13,15)(3,14,7,4)(10,12,16,11)Polycyclic presentation:
2^9|b^2=i,c^2=i,d^2=i,e^2=i,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=dg,e^a=ei,f^a=fi,g^a=gi,h^a=hi,c^b=ci,e^b=ei,f^b=fhi,e^c=eh,f^c=fi,g^d=gi,f^e=fi53 conjugacy classes — view the full character table.
| maximal subgroup of | #72209905, #72209909, #72234102, … |
| maximal normal subgroup of | #72209905, #72234102, #72260545, … |
| maximal quotient of | #72213362, #72227828, #72228003, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 419034);