Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #13657 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42 · Schur multiplier (H2) = C22×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2177 |
| commutator | C16 | C2×C42 |
| Frattini, cosocle | C22×C16 | C23 |
| prime radical | C2×#508 | C22 |
| LCS | G ▷ C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ C2×C4×C8 ◁ C2×#346 ◁ G | |
| LECS | G ▷ C22×C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
288 conjugacy classes of subgroups; only the 106 normal subgroups are shown.
This group has 288 conjugacy classes of subgroups — too many to draw. Only its 106 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 3 generators on 36 points:
(1,2)(3,4)(5,6,10,16,13,19,12,34)(7,20,22,27,24,31,17,9)(8,14,26,35,29,36,28,18)(11,32,30,21,33,23,25,15);(2,4)(5,7,11,14,8,23,27,6,10,17,30,18,26,21,31,34,13,24,33,36,29,32,9,19,12,22,25,35,28,15,20,16);(5,8,10,26,13,29,12,28)(6,14,16,35,19,36,34,18)(7,21,22,23,24,15,17,32)(9,30,20,33,27,25,31,11)Polycyclic presentation:
2^9|a^2=e,b^2=f,c^2=e,d^2=ghi,e^2=i,f^2=g,g^2=hi,h^2=i,b^a=bd,d^a=dg,f^a=fghi,g^a=gh,h^a=hi,c^b=ch,e^b=ei,d^c=di,f^c=fi68 conjugacy classes — view the full character table.
| maximal subgroup of | C4×#13658, #72223645, C2×#151972, … |
| maximal normal subgroup of | C4×#13658, #72223645, C2×#151972, … |
| maximal quotient of | #72223668, #72223678, C2×#151972, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 59168);