Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #511771 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×B(4,𝔽2) |
| commutator, Frattini, cosocle | C22×D4 | C24 |
| prime radical | #23906 | C2 |
| LCS | G ▷ C22×D4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C24 ◁ C23×D4 ◁ G | |
| LECS | G ▷ C22×D4 ▷ C22 ▷ C2 ▷ C1 | |
2784 conjugacy classes of subgroups; only the 192 normal subgroups are shown.
This group has 2784 conjugacy classes of subgroups — too many to draw. Only its 192 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) = 32 |
A minimal faithful permutation representation — 4 generators on 20 points:
(1,2)(3,4)(5,6)(7,15)(8,12)(9,13)(10,14)(11,16)(17,19)(18,20);(5,7,10,8)(6,11,14,19)(9,16,18,17)(12,13,15,20);(5,8,10,7)(6,12,14,15)(9,16,18,17)(11,13,19,20);(2,4)Polycyclic presentation:
2^9|b^2=i,c^2=i,b^a=be,c^a=cf,d^a=dg,h^a=hi,f^b=fhi,e^c=eh,f^e=fi65 conjugacy classes — view the full character table.
| maximal subgroup of | #72237595, #72237597, #72251941, … |
| maximal normal subgroup of | #72237595, #72237597, #72251941, … |
| maximal quotient of | C2×#511773, #72279556, #72281107, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 418969);