Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #244791 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | D42 |
| commutator, Frattini, cosocle | C2×C42 | C24 |
| LCS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
1884 conjugacy classes of subgroups; only the 202 normal subgroups are shown.
This group has 1884 conjugacy classes of subgroups — too many to draw. Only its 202 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) = 128 |
A minimal faithful permutation representation — 4 generators on 20 points:
(1,2)(3,4)(5,6)(7,9)(8,11)(10,12)(13,14)(15,17)(16,19)(18,20);(6,9)(7,12)(10,11)(13,14)(15,17)(16,19)(18,20);(2,4)(14,17)(15,20)(18,19);(6,9)(7,12)(10,11)Polycyclic presentation:
2^9|e^2=h,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=de,e^a=eh,f^a=fi,g^a=gi,c^b=cg,e^b=eh,f^b=fi,g^b=gi,f^c=fi,g^c=gi,e^d=eh71 conjugacy classes — view the full character table.
| maximal subgroup of | #72310284, #72310289, #72310301, … |
| maximal normal subgroup of | #72310284, #72310289, #72310301, … |
| maximal quotient of | C2×#244791 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 151987);