Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #213828 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2308 |
| commutator | C24 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| prime radical | C22×H(5,𝔽2) | C22 |
| LCS | G ▷ C24 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C22 ▷ C1 | |
1772 conjugacy classes of subgroups; only the 206 normal subgroups are shown.
This group has 1772 conjugacy classes of subgroups — too many to draw. Only its 206 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)(3,7,8,5)(4,6)(9,10,13,18,14,17,26,32)(11,19,22,33,21,15,25,28)(12,16,23,31,24,30,37,40)(20,27,35,38,34,29,36,39);(9,11)(10,15)(12,20)(13,25)(14,21)(16,29)(17,19)(18,33)(22,26)(23,36)(24,34)(27,30)(28,32)(31,38)(35,37)(39,40);(1,3)(2,5)(4,8)(6,7)(9,12,14,24)(10,16,17,30)(11,20,21,34)(13,23,26,37)(15,27,19,29)(18,31,32,40)(22,35,25,36)(28,38,33,39);(10,17)(13,26)(15,19)(16,30)(22,25)(23,37)(27,29)(35,36)Polycyclic presentation:
2^9|a^2=g,c^2=h,g^2=h,b^a=be,c^a=cf,d^a=de,e^a=eh,f^a=fi,c^b=ce,f^b=fh,g^b=gh,g^c=gi,g^d=gh56 conjugacy classes — view the full character table.
| maximal subgroup of | #72310752, #72310753, #72310760, … |
| maximal normal subgroup of | #72310752, #72310753, #72310760, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 121087);