Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #476654 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C22×C42 |
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 | |
752 conjugacy classes of subgroups; only the 154 normal subgroups are shown.
This group has 752 conjugacy classes of subgroups — too many to draw. Only its 154 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:
(1,2,4,6)(3,5,7,8)(9,10,13,16,15,20,14,19)(11,21,23,17,25,31,24,33)(12,18,27,34,29,36,28,30)(22,37,26,40,39,35,38,32);(1,3)(2,5)(4,7)(6,8)(10,16,20,19)(17,31,33,21)(18,34,36,30)(32,37,40,35);(3,7)(5,8)(9,11,15,25)(10,17,20,33)(12,26,29,38)(13,24,14,23)(16,21,19,31)(18,35,36,37)(22,28,39,27)(30,32,34,40);(9,12)(10,18,20,36)(11,22)(13,28)(14,27)(15,29)(16,30,19,34)(17,32,33,40)(21,37,31,35)(23,38)(24,26)(25,39)Polycyclic presentation:
2^9|a^2=eg,b^2=ef,c^2=i,d^2=ef,e^2=i,f^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,g^a=gi,h^a=hi,c^b=cg,d^b=def,d^c=de,e^c=ei,f^c=fi,e^d=ei,f^d=fi,h^d=hi50 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 386502);