Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #485343 |
| 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 | #18350 | C2 |
| LCS | G ▷ C2×C4⋊C4 ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ C2×C4×D4 ◁ G | |
| LECS | G ▷ C2×C4⋊C4 ▷ C23 ▷ C2 ▷ C1 | |
836 conjugacy classes of subgroups; only the 118 normal subgroups are shown.
This group has 836 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)(11,15)(12,22)(13,26)(14,18)(16,25)(17,27)(19,34)(20,33)(21,30)(23,38)(24,37)(28,35)(29,36)(31,39)(32,40);(1,2)(3,6)(4,8)(5,7)(9,11,14,21)(10,15,18,30)(12,23,24,19)(13,20,27,36)(16,31,32,28)(17,33,26,29)(22,34,37,38)(25,39,40,35);(9,12,14,24)(10,16)(11,19,21,23)(13,25,27,40)(15,28)(17,22)(18,32)(20,35,36,39)(26,37)(29,34)(30,31)(33,38);(1,3,5,4)(2,6,7,8)(12,24)(16,32)(19,23)(22,37)(25,40)(28,31)(34,38)(35,39)Polycyclic presentation:
2^9|b^2=i,c^2=h,d^2=g,e^2=g,f^2=i,b^a=be,c^a=cf,d^a=dg,e^a=eg,f^a=fi,h^a=hi,c^b=ci,e^b=eg,f^b=fhi,d^c=di,e^c=eh,f^e=fi41 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 392668);