Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #30056 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C22×C42 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2631 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| prime radical | C22×AGL(1,ℤ/8) | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1029 conjugacy classes of subgroups.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 20 points:
(1,2,3,4)(5,6,10,8,11,14,9,13)(7,15,16,12,18,20,17,19);(5,7)(6,12)(8,15)(9,16)(10,17)(11,18)(13,20)(14,19);(6,8)(7,16)(9,10)(12,19)(13,14)(17,18);(5,8,9,6,11,13,10,14)(7,15,16,12,18,20,17,19)Polycyclic presentation:
2^9|a^2=h,d^2=f,e^2=i,f^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,g^a=gi,c^b=cg,e^b=ei,g^b=gi,h^b=hi,d^c=de,e^c=ei,f^c=fi,g^c=gi,h^c=hi,g^d=gi68 conjugacy classes — view the full character table.
| maximal subgroup of | #72224752, #72224755, #72224757 |
| maximal normal subgroup of | #72224752, #72224755, #72224757 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 287626);