Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #486139 |
| 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 | C2×C22⋊C4 | C24 |
| prime radical | #58997 | C2 |
| LCS | G ▷ C2×C22⋊C4 ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ C2×C4×D4 ◁ G | |
| LECS | G ▷ C2×C22⋊C4 ▷ C23 ▷ C2 ▷ C1 | |
1611 conjugacy classes of subgroups; only the 134 normal subgroups are shown.
This group has 1611 conjugacy classes of subgroups — too many to draw. Only its 134 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 16 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 16 points:
(1,2,5,8)(3,7,4,6)(9,10)(11,13)(12,14)(15,16);(2,6)(3,4)(7,8)(10,13)(14,16);(11,15);(1,3,5,4)(2,6,8,7)Polycyclic presentation:
2^9|a^2=g,d^2=g,e^2=g,b^a=be,c^a=cf,d^a=dg,e^a=eg,h^a=hi,d^b=dg,e^b=eg,f^b=fhi,e^c=eh,f^e=fi53 conjugacy classes — view the full character table.
| maximal subgroup of | #72231271, #72231294, #72281868, … |
| maximal normal subgroup of | #72231271, #72231294, #72281868, … |
| maximal quotient of | #72282512, #72282976, #72283108, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 393543);