Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #142216 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | B(4,𝔽2) |
| commutator, Frattini, cosocle | C22×C22⋊C4 | C23 |
| prime radical | C2×#1595 | C2 |
| LCS | G ▷ C22×C22⋊C4 ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C24 ◁ C22×C22⋊C4 ◁ G | |
| LECS | G ▷ C22×C22⋊C4 ▷ C23 ▷ C2 ▷ C1 | |
1047 conjugacy classes of subgroups; only the 78 normal subgroups are shown.
This group has 1047 conjugacy classes of subgroups — too many to draw. Only its 78 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 — 3 generators on 20 points:
(1,2)(3,4)(5,6)(7,11)(8,10)(9,12)(14,17)(18,20);(6,10)(7,8)(11,12)(13,14)(15,17)(16,18)(19,20);(2,4)(5,7,9,8)(6,11,12,10)(15,19)Polycyclic presentation:
2^9|c^2=g,d^2=g,b^a=bd,c^a=ce,d^a=dg,f^a=fhi,c^b=cf,d^b=dg,h^b=hi,d^c=dh,f^d=fi50 conjugacy classes — view the full character table.
| maximal subgroup of | #72240328, #72240371, #72240372 |
| maximal normal subgroup of | #72240328, #72240371, #72240372 |
| maximal quotient of | #72261483 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 49460);