Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #485597 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×B(4,𝔽2) |
| commutator, Frattini, cosocle | C2×C22⋊C4 | C24 |
| prime radical | #2404 | C22 |
| LCS | G ▷ C2×C22⋊C4 ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C24 ◁ C22×C22⋊C4 ◁ G | |
| LECS | G ▷ C2×C22⋊C4 ▷ C23 ▷ C2 ▷ C1 | |
950 conjugacy classes of subgroups; only the 122 normal subgroups are shown.
This group has 950 conjugacy classes of subgroups — too many to draw. Only its 122 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2,5,8)(3,9,10,16)(4,12,14,7)(6,13,15,11)(17,18)(19,23)(20,22)(21,24)(25,29)(26,31)(27,32)(28,30);(1,3)(2,6)(4,13)(5,10)(7,16)(8,15)(9,12)(11,14)(18,22)(23,29)(24,30)(31,32);(17,19,21,26)(18,23,24,31)(20,27,28,25)(22,29,30,32);(3,10)(6,15)(9,16)(11,13)Polycyclic presentation:
2^9|a^2=g,c^2=i,e^2=g,b^a=be,c^a=cf,e^a=eg,h^a=hi,d^b=dg,e^b=eg,f^b=fhi,e^c=eh,f^e=fi50 conjugacy classes — view the full character table.
| maximal subgroup of | #72284337, #72285734, #72285789 |
| maximal normal subgroup of | #72284337, #72285734, #72285789 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 392875);