Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #484212 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×#447 |
| commutator, Frattini, cosocle | C22×C8 | C24 |
| LCS | G ▷ C22×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ C2×#408 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
1165 conjugacy classes of subgroups; only the 138 normal subgroups are shown.
This group has 1165 conjugacy classes of subgroups — too many to draw. Only its 138 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:
(5,8)(9,10)(11,22)(12,16)(13,21)(14,20)(15,18)(17,24)(19,26)(23,36)(25,37)(27,35)(28,33)(29,32)(30,34)(31,40)(38,39);(1,2)(3,5)(4,6)(7,8)(10,16)(12,26)(13,14)(17,31)(18,32)(19,21)(20,33)(22,36)(23,39)(24,30)(27,40)(28,29)(34,38)(35,37);(1,3)(2,5)(4,7)(6,8)(9,11)(10,17)(12,27)(13,30)(14,24)(15,25)(16,31)(18,34)(19,36)(20,37)(21,22)(23,29)(26,40)(28,39)(32,38)(33,35);(10,18)(16,32)(17,34)(19,33)(20,21)(22,37)(31,38)(35,36)Polycyclic presentation:
2^9|e^2=gi,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=di,e^a=eg,f^a=fi,g^a=gi,e^b=eg,f^b=fh,g^b=gi,e^c=eh,f^c=fi50 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 391408);