Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #417903 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C23×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2684 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C24 | H(5,𝔽2) |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
896 conjugacy classes of subgroups; only the 170 normal subgroups are shown.
This group has 896 conjugacy classes of subgroups — too many to draw. Only its 170 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)(9,10,14,19)(11,20,23,31)(12,24)(13,15,29,18)(16,27,33,22)(17,28)(21,36)(25,34)(26,30)(32,38)(35,39)(37,40);(2,5)(3,7)(10,15,19,18)(12,25,26,28)(16,31,33,20)(17,34)(21,37,38,39)(24,30)(32,36)(35,40);(9,11,14,23)(10,16,19,33)(12,21,26,38)(13,27,29,22)(15,20,18,31)(17,32,34,36)(24,40,30,35)(25,39,28,37);(1,3,4,7)(2,6,5,8)(9,12,14,26)(10,17,19,34)(11,21,23,38)(13,28,29,25)(15,30,18,24)(16,32,33,36)(20,35,31,40)(22,39,27,37)Polycyclic presentation:
2^9|a^2=fgh,b^2=g,c^2=i,d^2=ef,e^2=i,f^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,g^a=gi,c^b=cg,d^b=de,e^c=ei,f^c=fi,h^c=hi,e^d=ei,f^d=fi,g^d=gi,h^d=hi62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 360175);