Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #486795 |
| 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 | C2×#1595 | C2 |
| LCS | G ▷ C2×C22⋊C4 ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C24 ◁ C22×C22⋊C4 ◁ G | |
| LECS | G ▷ C2×C22⋊C4 ▷ C23 ▷ C2 ▷ C1 | |
1091 conjugacy classes of subgroups; only the 118 normal subgroups are shown.
This group has 1091 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) = 32 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2)(3,8)(4,7)(5,10)(6,11)(9,16)(12,22)(13,20)(14,24)(15,25)(17,31)(18,21)(19,27)(23,29)(26,30)(28,32);(2,7)(4,16)(8,20)(9,10)(11,21)(13,29)(17,28)(18,30)(22,32)(23,24)(25,31)(26,27);(1,3,6,15)(2,8,11,25)(4,17,18,13)(5,14,19,12)(7,20,21,31)(9,23,26,32)(10,24,27,22)(16,28,30,29);(3,12)(8,22)(13,28)(14,15)(17,29)(20,32)(23,31)(24,25)Polycyclic presentation:
2^9|c^2=i,e^2=g,b^a=be,c^a=cf,e^a=eg,h^a=hi,e^b=eg,f^b=fhi,d^c=dgi,e^c=eh,f^e=fi47 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 393993);