Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #483956 |
| 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 | C23 | C2×AGL(1,ℤ/8) |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ C4×#97 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
557 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 40 points:
(1,2)(3,6)(4,5)(7,8)(9,10,15,21)(11,23)(12,29)(13,32,14,35)(16,36,33,22)(17,27)(18,30)(19,24,20,34)(25,40)(26,39)(28,38)(31,37);(2,5)(6,8)(9,11,14,27,16,12,24,31,15,28,13,26,33,25,34,30)(10,17,20,38,22,18,35,29,21,39,19,23,36,37,32,40);(1,3,4,7)(2,6,5,8)(9,12,13,30,16,28,34,27,15,25,14,31,33,11,24,26)(10,18,19,40,22,39,32,38,21,37,20,29,36,17,35,23);(11,25,28,12)(13,14)(17,37,39,18)(19,20)(23,40,38,29)(24,34)(26,30,27,31)(32,35)Polycyclic presentation:
2^9|a^2=g,b^2=f,c^2=e,d^2=gi,e^2=h,f^2=h,h^2=i,b^a=be,c^a=cf,e^a=eh,f^a=fh,h^a=hi,d^b=dh,g^b=gi,d^c=dh,g^c=gi,e^d=ei,f^d=fi68 conjugacy classes — view the full character table.
| maximal subgroup of | #72309598, #72309632, #72309641, … |
| maximal normal subgroup of | #72309598, #72309632, #72309641, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 403256);