Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #124826 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C2×#99 |
| commutator | C23×C4 | C22×C4 |
| Frattini, cosocle | C22×C22⋊C4 | C23 |
| prime radical | C2×#2404 | C2 |
| LCS | G ▷ C23×C4 ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C25 ◁ C23×C22⋊C4 ◁ G | |
| LECS | G ▷ C22×C22⋊C4 ▷ C23 ▷ C2 ▷ C1 | |
2090 conjugacy classes of subgroups; only the 108 normal subgroups are shown.
This group has 2090 conjugacy classes of subgroups — too many to draw. Only its 108 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 3 generators on 20 points:
(1,2)(3,4)(6,10,11,12)(13,14)(15,19,20,17)(16,18);(5,6)(7,10)(8,12)(9,11)(14,17)(18,19);(2,4)(5,7,9,8)(6,10,11,12)(13,15)(14,17)(16,20)(18,19)Polycyclic presentation:
2^9|a^2=f,c^2=h,d^2=h,b^a=bd,c^a=ce,d^a=dg,e^a=ei,g^a=gi,d^b=dh,f^b=fgh,f^c=fi,f^d=fi62 conjugacy classes — view the full character table.
| maximal subgroup of | #72268782, #72283946, #72284431, … |
| maximal normal subgroup of | #72268782, #72283946, #72284431, … |
| maximal quotient of | #72268846, #72268848, #72284187, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 37065);