Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #17577 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C42 · Schur multiplier (H2) = C25×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C22×C22⋊C4 |
| commutator | C2×C4 | C22×C42 |
| Frattini, cosocle | C2×C42 | C24 |
| socle | C22 | C22×#95 |
| prime radical | C25 | C22⋊C4 |
| LCS | G ▷ C2×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C23×C42 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
1722 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 24 points:
(1,2,6,8)(3,5,7,4)(9,10,13,17,14,18,12,16)(11,19,20,15,22,23,21,24);(1,3,5,8,6,7,4,2)(9,11)(10,15)(12,21)(13,20)(14,22)(16,19)(17,23)(18,24);(1,4,6,5)(2,3,8,7)(9,12,14,13)(10,16,18,17)(11,20,22,21)(15,23,24,19);(1,4,6,5)(2,7,8,3)(9,12,14,13)(10,17,18,16)(11,21,22,20)(15,23,24,19)Polycyclic presentation:
2^9|a^2=f,b^2=g,c^2=h,d^2=h,e^2=h,f^2=i,g^2=hi,b^a=be,d^a=dh,e^a=eh,g^a=ghi,c^b=ch,e^b=ei104 conjugacy classes — view the full character table.
| maximal subgroup of | #72239003, #72239005, #72239008, … |
| maximal normal subgroup of | #72239003, #72239005, #72239008, … |
| maximal quotient of | #72257790 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 88997);