Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #508078 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×B(4,𝔽2) |
| commutator, Frattini, cosocle | C22×Q8 | C24 |
| prime radical | #3487 | C22 |
| LCS | G ▷ C22×Q8 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ C22×#41 ◁ G | |
| LECS | G ▷ C22×Q8 ▷ C22 ▷ C2 ▷ C1 | |
1048 conjugacy classes of subgroups; only the 124 normal subgroups are shown.
This group has 1048 conjugacy classes of subgroups — too many to draw. Only its 124 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)(9,10,16,11)(12,23,27,35)(13,20,15,18)(14,22,30,19)(17,29,21,26)(24,37,33,40)(25,39,31,38)(28,34,32,36);(2,5)(6,8)(9,11,16,10)(12,24,27,33)(13,18,15,20)(14,22,30,19)(17,25,21,31)(23,37,35,40)(26,38,29,39)(28,34,32,36);(1,3)(2,6)(4,7)(5,8)(9,12)(10,17)(11,21)(13,25)(14,29)(15,31)(16,27)(18,33)(19,35)(20,24)(22,23)(26,30)(28,37)(32,40)(34,39)(36,38);(3,7)(6,8)(9,13,16,15)(10,18,11,20)(12,25,27,31)(14,28,30,32)(17,33,21,24)(19,34,22,36)(23,38,35,39)(26,40,29,37)Polycyclic presentation:
2^9|a^2=i,b^2=i,d^2=i,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=di,f^a=fi,g^a=ghi,c^b=cg,d^b=di,f^b=fh,g^b=gi,d^c=de,f^c=fi,g^c=gi,h^c=hi,g^f=gi41 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 415281);