Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #384421 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #14379 |
| 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 ◁ C2 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
988 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2,6,8,7,12,5,11)(3,13,14,24,17,23,16,9)(4,10,18,25,19,26,20,21)(15,27,29,22,30,31,28,32);(2,8)(3,14)(4,18)(5,6)(11,12)(13,23)(16,17)(19,20)(21,25)(22,31)(27,32)(28,29);(1,3,7,17)(2,9)(4,15,19,30)(5,16,6,14)(8,13)(10,22)(11,23)(12,24)(18,29,20,28)(21,27)(25,31)(26,32);(1,4,5,18,7,19,6,20)(2,10,11,25,12,26,8,21)(3,15,16,29,17,30,14,28)(9,22,23,31,24,32,13,27)Polycyclic presentation:
2^9|a^2=h,c^2=ef,d^2=e,e^2=i,f^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=def,f^a=fi,g^a=gi,c^b=cg,d^b=de,e^b=ei,f^b=fi,g^b=gi,h^b=hi,g^c=gi,h^d=hi65 conjugacy classes — view the full character table.
| maximal subgroup of | #72251689, #72251695, #72251699, … |
| maximal normal subgroup of | #72251689, #72251695, #72251699, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 291619);