Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #486097 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C23×C42 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×B(4,𝔽2) |
| commutator, Frattini, cosocle | C2×C22⋊C4 | C24 |
| LCS | G ▷ C2×C22⋊C4 ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ C2×#117 ◁ G | |
| LECS | G ▷ C2×C22⋊C4 ▷ C23 ▷ C2 ▷ C1 | |
1037 conjugacy classes of subgroups; only the 118 normal subgroups are shown.
This group has 1037 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)(3,12)(4,7)(5,10)(6,11)(8,16)(9,17)(13,23)(14,28)(15,24)(18,21)(19,26)(20,31)(22,30)(25,32)(27,29);(2,7)(3,13)(4,17)(8,22)(9,10)(11,21)(12,27)(14,16)(15,30)(18,32)(20,28)(23,24)(25,26)(29,31);(3,14)(4,18)(8,23)(12,28)(13,16)(17,32)(20,27)(22,24);(1,3,6,14)(2,8,11,23)(4,15,18,30)(5,16,19,13)(7,20,21,27)(9,24,25,22)(10,12,26,28)(17,31,32,29)Polycyclic presentation:
2^9|d^2=i,e^2=g,b^a=be,c^a=cf,d^a=dg,e^a=eg,h^a=hi,d^b=dgi,e^b=eg,f^b=fhi,d^c=di,e^c=eh,f^e=fi47 conjugacy classes — view the full character table.
| maximal subgroup of | #72338699, #72338739, #72341009 |
| maximal normal subgroup of | #72338699, #72338739, #72341009 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 393294);