Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #486859 |
| 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×C4⋊C4 | C24 |
| prime radical | #52410 | C2 |
| LCS | G ▷ C2×C4⋊C4 ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C24 ◁ C22×C4⋊C4 ◁ G | |
| LECS | G ▷ C2×C4⋊C4 ▷ C23 ▷ C2 ▷ C1 | |
973 conjugacy classes of subgroups; only the 118 normal subgroups are shown.
This group has 973 conjugacy classes of subgroups — too many to draw. Only its 118 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:
(1,2,5,9)(3,7,10,14)(4,6,12,8)(11,13,16,15)(17,18)(19,25)(20,28)(21,24)(22,27)(23,30)(26,32)(29,31);(2,6)(4,12)(7,13)(8,9)(11,16)(14,15)(23,28)(25,32);(1,3)(2,7)(4,11)(5,10)(6,13)(8,15)(9,14)(12,16)(17,19)(18,22)(20,29)(21,26)(23,32)(24,31)(25,28)(27,30);(3,10)(7,14)(11,16)(13,15)(19,26)(22,31)(25,32)(27,29)Polycyclic presentation:
2^9|a^2=g,e^2=g,f^2=i,b^a=be,c^a=cf,e^a=eg,f^a=fi,h^a=hi,e^b=eg,f^b=fhi,d^c=dgi,e^c=eh,f^c=fi,f^e=fi41 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 394183);