Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #12077 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C23×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #442 |
| commutator | C2×C8 | C23×C4 |
| Frattini, cosocle | C22×C8 | C24 |
| socle | C22 | C2×#442 |
| prime radical | C24 | AGL(1,ℤ/8) |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C23×C4 ◁ C22×#97 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
606 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)(11,23)(12,27,30,40)(13,19)(14,31)(15,20)(16,22)(17,25)(18,24,34,39)(21,26)(28,36,29,37)(32,33,38,35);(2,5)(6,8)(9,11,15,14,16,26,13,25)(10,17,19,21,22,31,20,23)(12,24,29,32,30,39,28,38)(18,27,33,36,34,40,35,37);(1,3,4,7)(2,6,5,8)(9,12)(10,18)(11,24)(13,28)(14,32)(15,29)(16,30)(17,27)(19,33)(20,35)(21,36)(22,34)(23,37)(25,38)(26,39)(31,40);(9,13,16,15)(10,19,22,20)(11,25,26,14)(12,28,30,29)(17,21,31,23)(18,33,34,35)(24,38,39,32)(27,36,40,37)Polycyclic presentation:
2^9|a^2=g,b^2=gh,c^2=eh,d^2=i,e^2=i,f^2=h,h^2=i,b^a=be,c^a=cf,e^a=ei,f^a=fh,h^a=hi,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 | #72264774, #72264806, #72264874, … |
| maximal normal subgroup of | #72264774, #72264806, #72264874, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 402007);