Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #331007 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C23×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2631 |
| commutator, Frattini, cosocle | C2×C42 | C24 |
| prime radical | #10218 | C2 |
| LCS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
604 conjugacy classes of subgroups; only the 124 normal subgroups are shown.
This group has 604 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) = 128 |
A minimal faithful permutation representation — 4 generators on 40 points:
(9,10,13,18,14,19,27,16)(11,20,22,30,24,28,23,15)(12,17,25,32,26,33,38,31)(21,34,35,40,37,39,36,29);(1,2)(3,5)(4,7)(6,8)(9,11)(10,15)(12,21)(13,23)(14,24)(16,20)(17,29)(18,28)(19,30)(22,27)(25,36)(26,37)(31,34)(32,39)(33,40)(35,38);(2,5)(3,8)(6,7)(10,16)(12,25)(13,27)(15,28)(18,19)(20,30)(21,35)(22,23)(26,38)(31,32)(34,39)(36,37);(9,12,13,25,14,26,27,38)(10,17,18,32,19,33,16,31)(11,21,23,36,24,37,22,35)(15,29,28,39,30,40,20,34)Polycyclic presentation:
2^9|a^2=e,d^2=f,e^2=h,f^2=h,g^2=i,b^a=be,c^a=cf,g^a=gh,c^b=cg,e^b=eh,g^b=gi,d^c=de,e^c=eh,f^c=fh,g^c=gi,g^d=gh50 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 238203);