Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #484587 |
| 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 | #18122 | C2 |
| LCS | G ▷ C2×C22⋊C4 ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ C2×#117 ◁ G | |
| LECS | G ▷ C2×C22⋊C4 ▷ C23 ▷ C2 ▷ C1 | |
1004 conjugacy classes of subgroups; only the 130 normal subgroups are shown.
This group has 1004 conjugacy classes of subgroups — too many to draw. Only its 130 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 40 points:
(2,6)(3,4)(7,8)(9,10)(11,19)(12,17)(13,15)(14,18)(16,22)(20,32)(21,34)(23,31)(24,29)(25,33)(26,28)(27,30)(35,39)(36,38)(37,40);(1,2)(3,6)(4,8)(5,7)(10,15)(16,28)(17,29)(18,30)(19,34)(31,38)(32,39)(33,40);(1,3,5,4)(2,6,7,8)(9,11)(10,16,18,19)(12,23)(13,26)(14,22)(15,28,30,34)(17,32,33,31)(20,25)(21,27)(24,35)(29,39,40,38)(36,37);(9,12,14,25)(10,17,18,33)(11,20,22,23)(13,24,27,37)(15,29,30,40)(16,31,19,32)(21,35,26,36)(28,38,34,39)Polycyclic presentation:
2^9|c^2=ghi,d^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=di,e^c=eh,f^c=fi,f^e=fi50 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 391978);