Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #484095 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×#442 |
| commutator | C2×C8 | C23×C4 |
| Frattini, cosocle | C4×C8 | C24 |
| prime radical | C23 | #442 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ #1451 ◁ G | |
| LECS | G ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
427 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 4 generators on 40 points:
(1,2,4,7,5,3,6,8)(9,10)(11,21,23,32)(12,25)(13,17)(14,29)(15,20)(16,22,33,39)(18,26)(19,31)(24,36,30,38)(27,37,40,34)(28,35);(9,11,15,23)(10,16,20,33)(12,22,26,39)(13,27,28,40)(14,30,31,24)(17,32,35,21)(18,36,25,38)(19,37,29,34);(1,3,4,8,5,2,6,7)(9,12,14,13,15,26,31,28)(10,17,19,25,20,35,29,18)(11,22,30,27,23,39,24,40)(16,32,37,38,33,21,34,36);(2,3)(7,8)Polycyclic presentation:
2^9|a^2=f,b^2=i,c^2=fg,e^2=g,f^2=h,g^2=i,b^a=be,c^a=cg,d^a=dh,e^a=eg,g^a=gi,e^b=egi,f^b=fi,g^b=gi,d^c=dh,e^c=ei62 conjugacy classes — view the full character table.
| maximal subgroup of | #72321445, #72321460, #72321534, … |
| maximal normal subgroup of | #72321445, #72321460, #72321534, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 391295);