Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #508078 |
| 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 | |
738 conjugacy classes of subgroups; only the 124 normal subgroups are shown.
This group has 738 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:
(2,3)(6,7)(9,10,12,15)(11,19,21,34)(13,25,23,17)(14,28,24,18)(16,26,31,22)(20,38,29,37)(27,35,30,36)(32,40,33,39);(1,2)(3,5)(4,6)(7,8)(10,15)(11,20)(14,24)(16,32)(17,25)(19,37)(21,29)(22,40)(26,39)(27,30)(31,33)(34,38);(1,3,5,2)(4,6,8,7)(9,11,12,21)(10,16,15,31)(13,26,23,22)(14,29,24,20)(17,34,25,19)(18,32,28,33)(27,40,30,39)(35,38,36,37);(1,4,5,8)(2,6,3,7)(10,15)(11,21)(17,25)(18,28)(20,29)(22,26)(35,36)(39,40)Polycyclic presentation:
2^9|a^2=i,c^2=e,d^2=ei,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=di,f^a=fi,g^a=ghi,c^b=cg,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.
| maximal subgroup of | #72271412 |
| maximal normal subgroup of | #72271412 |
| maximal quotient of | #72271458 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 415442);