Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #249355 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2403 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | #35676 | C2 |
| LCS | G ▷ C23×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C22 ▷ C1 | |
1983 conjugacy classes of subgroups; only the 152 normal subgroups are shown.
This group has 1983 conjugacy classes of subgroups — too many to draw. Only its 152 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:
(6,7)(9,10)(11,15)(12,22)(13,25)(14,18)(16,24)(17,27)(19,34)(20,37)(21,30)(23,33)(26,32)(28,36)(29,39)(31,38)(35,40);(1,2,5,4)(3,6,8,7)(9,11,14,21)(10,15,18,30)(12,19,23,35)(13,20,27,39)(16,31,32,28)(17,29,25,37)(22,40,33,34)(24,36,26,38);(1,3)(2,6)(4,7)(5,8)(9,12,14,23)(10,16,18,32)(11,19,21,35)(13,26,27,24)(15,28,30,31)(17,33,25,22)(20,38,39,36)(29,40,37,34);(1,4,5,2)(3,7,8,6)(12,23)(13,27)(16,32)(17,25)(19,35)(20,39)(28,31)(29,37)Polycyclic presentation:
2^9|b^2=h,c^2=i,d^2=hi,f^2=i,b^a=be,c^a=cf,d^a=de,f^a=fi,g^a=gi,c^b=cg,f^b=fhi,d^c=di,e^c=eh,f^c=fi,f^d=fh50 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 156579);