Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #485337 |
| 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 | #24262 | C2 |
| LCS | G ▷ C2×C22⋊C4 ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ C2×C4×D4 ◁ G | |
| LECS | G ▷ C2×C22⋊C4 ▷ C23 ▷ C2 ▷ C1 | |
870 conjugacy classes of subgroups; only the 118 normal subgroups are shown.
This group has 870 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) = 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)(12,23)(13,26)(14,17)(15,22)(16,24)(18,29)(20,35)(21,33)(25,32)(27,31)(28,40)(30,38)(34,37)(36,39);(1,2)(3,6)(4,8)(5,7)(9,11,14,21)(10,15,17,31)(12,20,24,37)(13,27,29,22)(16,30,23,39)(18,33,26,19)(25,38,28,36)(32,35,40,34);(9,12,14,24)(10,16)(11,20,21,37)(13,28,29,25)(15,30)(17,23)(18,32)(19,34)(22,38,27,36)(26,40)(31,39)(33,35);(1,3,5,4)(2,6,7,8)(10,17)(11,21)(12,24)(18,26)(22,27)(25,28)(30,39)(34,35)Polycyclic presentation:
2^9|b^2=i,c^2=h,d^2=gi,e^2=g,b^a=be,c^a=cf,d^a=dg,e^a=eg,h^a=hi,d^b=di,e^b=eg,f^b=fhi,d^c=di,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, 392704);