Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #489347 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #18524 |
| commutator, Frattini, cosocle | C4×C8 | C24 |
| LCS | G ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C2×C4 ◁ C4×M16 ◁ G | |
| LECS | G ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
960 conjugacy classes of subgroups; only the 124 normal subgroups are shown.
This group has 960 conjugacy classes of subgroups — too many to draw. Only its 124 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:
(2,6)(3,10)(4,13)(7,11)(8,14)(9,15)(12,16);(3,10)(4,13)(6,11)(7,9)(12,16)(14,15);(1,2)(3,11)(4,7)(5,8)(6,10)(9,13)(12,15)(14,16);(3,12)(6,14)(10,16)(11,15)Polycyclic presentation:
2^9|e^2=g,f^2=hi,h^2=i,b^a=be,c^a=cf,d^a=dg,e^a=eg,f^a=fh,h^a=hi,e^b=eg,f^b=fh,h^b=hi,e^c=eh,f^c=fh,g^c=gi,h^c=hi,f^d=fi44 conjugacy classes — view the full character table.
| maximal subgroup of | #72214208, #72214826, C3×#489347, … |
| maximal normal subgroup of | #72214208, #72214826, C3×#489347, … |
| maximal quotient of | #72213969, #72214744, C3×#489347, … |
| holomorph of | D8, Q16 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 396543);