Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #511802 |
| 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 | C22×Q8 | C24 |
| prime radical | C22×D4 | C2×D4 |
| LCS | G ▷ C22×Q8 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ C2×C4×Q8 ◁ G | |
| LECS | G ▷ C22×Q8 ▷ C22 ▷ C2 ▷ C1 | |
875 conjugacy classes of subgroups; only the 152 normal subgroups are shown.
This group has 875 conjugacy classes of subgroups — too many to draw. Only its 152 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) = 128 |
A minimal faithful permutation representation — 4 generators on 40 points:
(1,2,4,6)(3,7,8,5)(9,10)(11,20)(12,24)(13,16)(14,19)(15,22)(17,27)(18,29)(21,35)(23,37)(25,31)(26,34)(28,38)(30,36)(32,39)(33,40);(1,3,4,8)(2,5,6,7)(11,21)(15,31)(20,35)(22,25)(23,28)(32,33)(37,38)(39,40);(9,11,14,21)(10,15,19,31)(12,25,27,22)(13,28,29,23)(16,32,18,33)(17,35,24,20)(26,39,30,40)(34,37,36,38);(10,16)(13,29)(15,32)(17,36)(18,19)(20,37)(23,28)(24,34)(26,30)(31,33)(35,38)(39,40)Polycyclic presentation:
2^9|a^2=e,b^2=e,c^2=i,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=dg,f^a=fi,g^a=gi,h^a=hi,c^b=ci,f^c=fi,g^c=ghi,f^d=fh,g^d=gi,g^f=gi53 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 419221);