Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #500839 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C22×D8 |
| commutator, Frattini, cosocle | C22×C8 | C24 |
| prime radical | C22×D16 | C22 |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C4×C4⋊C4 ◁ C8×C4⋊C4 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
604 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 4 generators on 28 points:
(5,6)(7,12)(8,10)(9,11)(13,14)(15,17)(16,26)(18,22)(19,28)(20,24)(21,23)(25,27);(1,2)(3,4)(5,7,9,8)(6,10,11,12)(13,15,18,23,19,16,27,24,20,25,26,28,21,22,17,14);(2,4)(5,8,9,7)(6,10,11,12)(13,14,17,22,21,28,26,25,20,24,27,16,19,23,18,15);(6,11)(10,12)(13,16,18,24,19,25,27,28,20,22,26,14,21,15,17,23)Polycyclic presentation:
2^9|b^2=eh,c^2=e,d^2=f,e^2=hi,f^2=h,h^2=i,b^a=be,c^a=cf,d^a=de,e^a=eh,f^a=fhi,h^a=hi,c^b=cg,d^b=di,d^c=di74 conjugacy classes — view the full character table.
| maximal subgroup of | #72319014, #72319542, #72319580 |
| maximal normal subgroup of | #72319014, #72319542, #72319580 |
| maximal quotient of | #72322595 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 411898);