Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #142220 |
| 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×C4⋊C4 | C23 |
| LCS | G ▷ C22×C4⋊C4 ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C24 ◁ C22×C4⋊C4 ◁ G | |
| LECS | G ▷ C22×C4⋊C4 ▷ C23 ▷ C2 ▷ C1 | |
1149 conjugacy classes of subgroups; only the 78 normal subgroups are shown.
This group has 1149 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)(7,9)(8,11)(10,12)(15,20);(6,9)(7,12)(10,11)(13,14)(15,17)(16,19)(18,20);(2,4)(14,17)(15,20)(18,19)Polycyclic presentation:
2^9|d^2=g,f^2=i,b^a=bd,c^a=ce,d^a=dg,f^a=fhi,c^b=cf,d^b=dg,f^b=fi,h^b=hi,d^c=dh,f^c=fi,f^d=fi50 conjugacy classes — view the full character table.
| maximal subgroup of | #72278608, #72278626, #72278647, … |
| maximal normal subgroup of | #72278608, #72278626, #72278647, … |
| maximal quotient of | #72338361, #72338389 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 49416);