Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #141751 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C2×C8 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #1957 |
| commutator | C4×C8 | C22×C4 |
| Frattini, cosocle | C2×C4×C8 | C23 |
| prime radical | #19603 | C2 |
| LCS | G ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ C2×C82 ◁ G | |
| LECS | G ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
359 conjugacy classes of subgroups; only the 98 normal subgroups are shown.
This group has 359 conjugacy classes of subgroups — too many to draw. Only its 98 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 3 generators on 32 points:
(1,2,4,7,6,10,5,9)(3,11,13,16,12,15,14,8)(17,18,22,28,23,29,24,27)(19,20,30,21,25,31,26,32);(2,7,10,9)(3,12)(8,15,16,11)(13,14)(17,19,21,29,22,26,31,28,23,25,32,18,24,30,20,27);(1,3,5,14,6,12,4,13)(2,8,7,11,10,16,9,15)(17,20,24,32,23,31,22,21)(18,25,28,26,29,19,27,30)Polycyclic presentation:
2^9|a^2=f,b^2=dfgi,c^2=ei,d^2=g,e^2=h,f^2=h,g^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=eh,g^a=gi,c^b=chi,d^b=di,f^b=fi68 conjugacy classes — view the full character table.
| maximal subgroup of | #72225286, #72225321, #72287582, … |
| maximal normal subgroup of | #72225286, #72225321, #72287582, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 49037);