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 | |
610 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)(9,10,13,16,15,20,12,18)(11,21,22,38,25,39,24,36)(14,28,27,37,29,35,26,19)(17,32,33,23,34,30,31,40);(2,5)(6,8)(10,16,20,18)(14,29)(17,31,34,33)(19,35,37,28)(21,38,39,36)(23,40)(26,27)(30,32);(1,3,4,7)(2,6,5,8)(9,11)(10,17)(12,22)(13,24)(14,30)(15,25)(16,31)(18,33)(19,36)(20,34)(21,35)(23,27)(26,40)(28,39)(29,32)(37,38);(9,12,15,13)(10,18,20,16)(11,22,25,24)(14,26,29,27)(17,33,34,31)(19,35,37,28)(21,38,39,36)(23,30,40,32)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,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 | #72239532, #72239548, #72239550, … |
| maximal normal subgroup of | #72239532, #72239548, #72239550, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 402011);