Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #484039 |
| 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 | C22×C8 | C24 |
| prime radical | C24 | AGL(1,ℤ/8) |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ #1752 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
536 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 40 points:
(2,5)(6,8)(9,10,15,21)(11,23,26,20)(12,27)(13,22,16,19)(14,31,25,17)(18,33)(24,36)(28,38)(29,37)(30,40)(32,34)(35,39);(1,2)(3,6)(4,5)(7,8)(9,11,16,14,15,26,13,25)(10,17,22,20,21,31,19,23)(12,24,28,32,30,33,29,39)(18,27,34,37,36,40,35,38);(1,3,4,7)(2,6,5,8)(9,12)(10,18)(11,24)(13,29)(14,32)(15,30)(16,28)(17,27)(19,35)(20,37)(21,36)(22,34)(23,38)(25,39)(26,33)(31,40);(2,5)(6,8)Polycyclic presentation:
2^9|a^2=g,b^2=ghi,c^2=eh,e^2=i,f^2=h,h^2=i,b^a=be,c^a=cf,e^a=ei,f^a=fh,h^a=hi,d^b=deh,f^b=fi,e^c=ei,f^c=fhi,g^c=gi,h^c=hi62 conjugacy classes — view the full character table.
| maximal subgroup of | #72264905, #72264923, #72265031, … |
| maximal normal subgroup of | #72264905, #72264923, #72265031, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 405428);