Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #105688 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42 · Schur multiplier (H2) = C22 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C4×C22⋊C4 |
| commutator | C42 | C2×C42 |
| Frattini, cosocle | C2×C4×C8 | C23 |
| socle | C22 | #1752 |
| prime radical | C2×#41 | M16 |
| LCS | G ▷ C42 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C4×M32 ◁ G | |
| LECS | G ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
304 conjugacy classes of subgroups; only the 110 normal subgroups are shown.
This group has 304 conjugacy classes of subgroups — too many to draw. Only its 110 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 3 generators on 24 points:
(1,2)(3,5)(4,7)(6,8)(9,10,13,19,14,16,22,23,15,11,17,20,12,18,21,24);(2,5,7,6)(10,16,11,18)(13,17)(19,24,20,23)(21,22);(9,11,13,20,14,18,22,24,15,10,17,19,12,16,21,23)Polycyclic presentation:
2^9|a^2=e,b^2=f,c^2=e,d^2=gi,e^2=h,h^2=i,b^a=bd,d^a=dg,f^a=fgi,c^b=ch,e^b=ei,f^c=fi86 conjugacy classes — view the full character table.
| maximal subgroup of | #72225551, #72227130, #72227159, … |
| maximal normal subgroup of | #72225551, #72227130, #72227159, … |
| maximal quotient of | #72225537, #72225546 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 35848);