Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #508066 |
| 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×D4 | C24 |
| prime radical | #3485 | C22 |
| LCS | G ▷ C22×D4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ C22×#41 ◁ G | |
| LECS | G ▷ C22×D4 ▷ C22 ▷ C2 ▷ C1 | |
1080 conjugacy classes of subgroups; only the 124 normal subgroups are shown.
This group has 1080 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) = 128 |
A minimal faithful permutation representation — 4 generators on 40 points:
(1,2)(3,6)(4,5)(7,8)(9,10,15,11)(12,22,25,32)(13,20,27,17)(14,21,29,18)(16,28,19,24)(23,36,38,35)(26,31,40,34)(30,37,33,39);(2,5)(6,8)(9,11,15,10)(12,19,25,16)(13,17,27,20)(14,18,29,21)(22,24,32,28)(23,30,38,33)(26,34,40,31)(35,39,36,37);(1,3)(2,6)(4,7)(5,8)(9,12)(10,16)(11,19)(13,23)(14,28)(15,25)(17,30)(18,32)(20,33)(21,22)(24,29)(26,39)(27,38)(31,36)(34,35)(37,40);(3,7)(6,8)(9,13,15,27)(10,17,11,20)(12,23,25,38)(14,26,29,40)(16,30,19,33)(18,31,21,34)(22,35,32,36)(24,37,28,39)Polycyclic presentation:
2^9|a^2=i,b^2=i,d^2=i,f^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,d^c=de,f^c=fi,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, 415299);