Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #391252 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C22×C42 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #14622 |
| 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 | |
951 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)(3,12)(4,17,21,26)(5,10)(6,11)(7,14)(8,20,27,19)(9,22)(13,23,18,25)(15,24)(16,28)(29,30,32,31);(1,3,5,14,6,16,22,15)(2,7,9,12,11,24,10,28)(4,18,20,30,21,13,19,31)(8,25,26,29,27,23,17,32);(1,4)(2,8)(3,13)(5,19)(6,21)(7,23)(9,17)(10,26)(11,27)(12,29)(14,30)(15,31)(16,18)(20,22)(24,25)(28,32);(4,19)(5,22)(7,24)(8,17)(9,10)(13,30)(14,15)(18,31)(20,21)(23,32)(25,29)(26,27)Polycyclic presentation:
2^9|a^2=h,b^2=e,e^2=i,f^2=i,b^a=be,c^a=cf,e^a=ei,c^b=cg,d^b=def,f^b=fi,g^b=gi,d^c=de,e^c=ei,f^c=fi,h^c=hi,e^d=ei,f^d=fi53 conjugacy classes — view the full character table.
| maximal subgroup of | #72251785, #72251790, #72251795, … |
| maximal normal subgroup of | #72251785, #72251790, #72251795, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 298448);