Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #486883 |
| 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 | C2×#537 | C22 |
| LCS | G ▷ C2×C4⋊C4 ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C24 ◁ C22×C4⋊C4 ◁ G | |
| LECS | G ▷ C2×C4⋊C4 ▷ C23 ▷ C2 ▷ C1 | |
782 conjugacy classes of subgroups; only the 118 normal subgroups are shown.
This group has 782 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) = 48 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 48 points:
(2,6)(4,11)(7,13)(8,12)(10,15)(14,16)(17,18)(19,23)(20,30)(21,34)(22,26)(24,33)(25,36)(27,44)(28,42)(29,39)(31,48)(32,43)(35,41)(37,45)(38,47)(40,46);(1,2,5,8)(3,7,9,14)(4,12,11,6)(10,16,15,13)(17,19,22,29)(18,23,26,39)(20,31,32,27)(21,28,36,47)(24,40,41,37)(25,42,34,38)(30,44,43,48)(33,46,35,45);(1,3)(2,7)(4,10)(5,9)(6,13)(8,14)(11,15)(12,16)(17,20,22,32)(18,24,26,41)(19,27,29,31)(21,35,36,33)(23,37,39,40)(25,43,34,30)(28,46,47,45)(38,48,42,44);(3,9)(7,14)(10,15)(13,16)Polycyclic presentation:
2^9|b^2=gi,c^2=i,e^2=g,f^2=i,b^a=be,c^a=cf,e^a=eg,f^a=fi,h^a=hi,c^b=ci,e^b=eg,f^b=fhi,d^c=dg,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, 394338);