Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #501084 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×B(4,𝔽2) |
| commutator, Frattini, cosocle | C22×D4 | C24 |
| prime radical | #23908 | C2 |
| LCS | G ▷ C22×D4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ C22×#41 ◁ G | |
| LECS | G ▷ C22×D4 ▷ C22 ▷ C2 ▷ C1 | |
1222 conjugacy classes of subgroups; only the 140 normal subgroups are shown.
This group has 1222 conjugacy classes of subgroups — too many to draw. Only its 140 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 — 4 generators on 20 points:
(5,6)(7,16)(8,12)(9,14,19,13)(10,15)(11,17,20,18);(1,2)(3,4)(5,7,10,8)(6,11,15,20)(9,18,19,17)(12,13,16,14);(2,4)(5,8,10,7)(6,12,15,16)(9,17,19,18)(11,13,20,14);(6,13)(9,19)(11,16)(12,20)(14,15)(17,18)Polycyclic presentation:
2^9|a^2=f,b^2=i,c^2=i,e^2=i,b^a=be,c^a=cf,d^a=de,e^a=eh,h^a=hi,c^b=cg,e^b=ei,f^b=fhi,e^c=eh,e^d=ei,f^d=fhi,f^e=fi53 conjugacy classes — view the full character table.
| maximal subgroup of | #72285418, #72302973, #72303224, … |
| maximal normal subgroup of | #72285418, #72302973, #72303224, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 408283);