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 | #23900 | C2 |
| LCS | G ▷ C22×D4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C24 ◁ C23×D4 ◁ G | |
| LECS | G ▷ C22×D4 ▷ C22 ▷ C2 ▷ C1 | |
3032 conjugacy classes of subgroups; only the 192 normal subgroups are shown.
This group has 3032 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)(7,13)(8,11)(9,12)(10,14)(15,19)(16,18)(17,20);(5,7,9,15)(6,10,12,17)(8,14,16,20)(11,13,18,19);(8,16)(14,20);(2,4)Polycyclic presentation:
2^9|b^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 | #72219040, #72231773, #72233906, … |
| maximal normal subgroup of | #72219040, #72231773, #72233906, … |
| maximal quotient of | #72234555, #72234963, #72237119, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 418968);