Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #251935 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2404 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C22×2−1+4 | C22 |
| LCS | G ▷ C23×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C22 ▷ C1 | |
1506 conjugacy classes of subgroups; only the 130 normal subgroups are shown.
This group has 1506 conjugacy classes of subgroups — too many to draw. Only its 130 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)(4,6)(7,8)(9,10,15,19)(11,16,22,13)(12,23,26,35)(14,29,32,18)(17,30,34,27)(20,37,24,40)(21,28,36,31)(25,38,33,39);(1,3)(2,5)(4,7)(6,8)(9,11)(10,16)(12,24,26,20)(13,19)(14,21)(15,22)(17,33,34,25)(18,31)(23,37,35,40)(27,39,30,38)(28,29)(32,36);(5,8)(9,12)(10,17)(11,20)(13,25)(14,30)(15,26)(16,33)(18,35)(19,34)(21,39)(22,24)(23,29)(27,32)(28,37)(31,40)(36,38);(9,13,15,16)(10,11,19,22)(12,25,26,33)(14,31,32,28)(17,20,34,24)(18,36,29,21)(23,39,35,38)(27,37,30,40)Polycyclic presentation:
2^9|a^2=h,b^2=e,d^2=h,f^2=h,b^a=be,c^a=cf,d^a=de,f^a=fh,g^a=ghi,c^b=cg,f^b=fi,g^b=gh,e^c=eh,f^c=fh,f^d=fh50 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 159137);