Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #483807 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C23×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×#442 |
| commutator | C2×C8 | C23×C4 |
| Frattini, cosocle | C22×C8 | C24 |
| prime radical | C24 | AGL(1,ℤ/8) |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ C2×#405 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
604 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 40 points:
(1,2)(3,6)(4,5)(7,8)(9,10,13,20,15,21,12,18)(11,22,23,16,26,34,25,36)(14,29,28,33,32,40,27,19)(17,37,38,24,35,31,39,30);(2,5)(6,8)(9,11)(10,16,21,36)(12,25)(13,23)(14,30)(15,26)(17,33,35,19)(18,22,20,34)(24,32)(27,31)(28,37)(29,38,40,39);(1,3,4,7)(2,6,5,8)(9,11)(10,17)(12,23)(13,25)(14,31)(15,26)(16,33)(18,38)(19,36)(20,39)(21,35)(22,40)(24,28)(27,30)(29,34)(32,37);(9,12,15,13)(10,18,21,20)(11,23,26,25)(14,27,32,28)(16,34,36,22)(17,38,35,39)(19,40,33,29)(24,31,30,37)Polycyclic presentation:
2^9|a^2=g,b^2=gh,c^2=eh,d^2=i,e^2=i,f^2=h,g^2=i,h^2=i,b^a=be,c^a=cf,e^a=ei,f^a=fh,h^a=hi,d^b=di,f^b=fi,e^c=ei,f^c=fhi,g^c=gi,h^c=hi80 conjugacy classes — view the full character table.
| maximal subgroup of | #72239529, #72239540, #72239548, … |
| maximal normal subgroup of | #72239529, #72239540, #72239548, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 402013);