Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #13331 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | #459 |
| commutator | C22×C8 | C22×C4 |
| Frattini, cosocle | C23×C8 | C23 |
| prime radical | C23×D8 | C22 |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C23×C4 ◁ C22×C4×C8 ◁ G | |
| LECS | G ▷ C23×C8 ▷ C4 ▷ C2 ▷ C1 | |
1200 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 3 generators on 40 points:
(1,2)(3,4)(9,10,14,21,15,20,32,22)(11,23,26,34,27,40,33,35)(12,19,13,18,30,39,29,17)(16,28,37,24,36,25,38,31);(2,4)(5,6)(7,8)(9,11,15,27)(10,16,20,36)(12,28,30,25)(13,31,29,24)(14,33,32,26)(17,35,18,34)(19,40,39,23)(21,38,22,37);(6,8)(10,17)(12,29)(13,30)(14,32)(16,34)(18,20)(19,22)(21,39)(23,37)(24,28)(25,31)(26,33)(35,36)(38,40)Polycyclic presentation:
2^9|a^2=g,b^2=i,d^2=h,e^2=hi,g^2=i,h^2=i,b^a=bd,c^a=ce,d^a=dh,e^a=ehi,f^a=fi,h^a=hi,c^b=cf,d^b=dhi,e^b=eh,g^b=gi,h^b=hi,d^c=dhi,e^c=eh,g^c=gi,h^c=hi80 conjugacy classes — view the full character table.
| maximal subgroup of | #72281717, #72281756, #72281975, … |
| maximal normal subgroup of | #72281717, #72281756, #72281975, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 55065);