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 | #29634 | 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 | |
999 conjugacy classes of subgroups; only the 118 normal subgroups are shown.
This group has 999 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)(3,7)(4,6)(5,9)(8,12)(10,14)(11,13)(15,16)(17,18)(19,25)(20,28)(21,24)(22,26)(23,30)(27,32)(29,31);(2,6)(4,12)(7,13)(8,9)(11,16)(14,15)(17,19,21,27)(18,22,24,31)(20,29,30,26)(23,32,28,25);(1,3)(2,7)(4,11)(5,10)(6,13)(8,15)(9,14)(12,16)(20,30)(26,29);(3,10)(7,14)(11,16)(13,15)Polycyclic presentation:
2^9|b^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, 393992);