Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #249391 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2403 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | #35463 | C2 |
| LCS | G ▷ C23×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C22 ▷ C1 | |
1531 conjugacy classes of subgroups; only the 152 normal subgroups are shown.
This group has 1531 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:
(1,2)(3,5)(4,7)(6,8)(9,10,14,18)(11,15,21,28)(12,16,23,30)(13,24,25,17)(19,31,33,26)(20,34,35,27)(22,29,36,37)(32,39,40,38);(3,8)(5,6)(9,11,14,21)(10,15,18,28)(12,19,23,33)(13,20,25,35)(16,26,30,31)(17,27,24,34)(22,32,36,40)(29,38,37,39);(2,5)(3,8)(6,7)(9,12)(10,16)(11,19)(13,22)(14,23)(15,26)(17,29)(18,30)(20,32)(21,33)(24,37)(25,36)(27,38)(28,31)(34,39)(35,40);(9,13)(10,17)(11,20)(12,22)(14,25)(15,27)(16,29)(18,24)(19,32)(21,35)(23,36)(26,38)(28,34)(30,37)(31,39)(33,40)Polycyclic presentation:
2^9|a^2=h,b^2=h,f^2=i,b^a=be,c^a=cf,d^a=deh,f^a=fi,g^a=gi,c^b=cg,f^b=fhi,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, 156589);