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 | C22×H(5,𝔽2) | C22 |
| LCS | G ▷ C23×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C22 ▷ C1 | |
2032 conjugacy classes of subgroups; only the 152 normal subgroups are shown.
This group has 2032 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:
(5,8)(9,10)(11,16)(12,23)(13,18)(14,27)(15,19)(17,25)(20,36)(21,30)(22,32)(24,34)(26,35)(28,31)(29,37)(33,39)(38,40);(1,2)(3,5)(4,6)(7,8)(9,11,15,22)(10,16,19,32)(12,20,26,38)(13,21,14,28)(17,33,34,29)(18,30,27,31)(23,36,35,40)(24,37,25,39);(1,3)(2,5)(4,7)(6,8)(9,12,15,26)(10,17,19,34)(11,20,22,38)(13,25,14,24)(16,29,32,33)(18,35,27,23)(21,39,28,37)(30,36,31,40);(1,2)(3,5)(4,6)(7,8)(9,13,15,14)(10,18,19,27)(11,21,22,28)(12,24,26,25)(16,30,32,31)(17,23,34,35)(20,37,38,39)(29,40,33,36)Polycyclic presentation:
2^9|b^2=i,c^2=i,d^2=i,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, 156581);