Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #499855 |
| 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 | C2×#535 | C22 |
| LCS | G ▷ C22×Q8 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ C22×#41 ◁ G | |
| LECS | G ▷ C22×Q8 ▷ C22 ▷ C2 ▷ C1 | |
1062 conjugacy classes of subgroups; only the 134 normal subgroups are shown.
This group has 1062 conjugacy classes of subgroups — too many to draw. Only its 134 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:
(1,2)(3,6)(4,5)(7,8)(9,10,15,19)(11,16,14,28)(12,22,23,35)(13,25,27,18)(17,26,34,24)(20,37,29,40)(21,36,30,32)(31,39,33,38);(2,5)(6,8)(9,11)(10,16)(12,23)(13,21)(14,15)(18,32)(19,28)(22,35)(25,36)(27,30)(31,33)(38,39);(1,3,4,7)(2,6,5,8)(9,12)(10,17)(11,20)(13,26)(14,29)(15,23)(16,31)(18,35)(19,34)(21,39)(22,25)(24,27)(28,33)(30,38)(32,40)(36,37);(2,5)(6,8)(9,11,15,14)(10,16,19,28)(12,20,23,29)(13,21,27,30)(17,31,34,33)(18,32,25,36)(22,37,35,40)(24,38,26,39)Polycyclic presentation:
2^9|a^2=i,c^2=e,d^2=i,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=de,f^a=fi,g^a=ghi,c^b=cg,d^b=di,f^b=fh,g^b=gi,f^c=fi,g^c=gi,h^c=hi,g^f=gi50 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 407400);