Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7518992 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | #25364 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C4×C4⋊Q8 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2099 conjugacy classes of subgroups.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 20 points:
(2,3)(5,6,9,8)(7,10,12,11)(13,14,17,18)(15,16,19,20);(1,2,4,3)(5,7)(6,10)(8,11)(9,12)(13,14,17,18)(15,16,19,20);(5,8,9,6)(7,11,12,10)(13,14,16,15,17,18,20,19);(7,12)(10,11)(13,15,16,18,17,19,20,14);(1,3,4,2)Polycyclic presentation:
2^9|a^2=gh,b^2=fi,c^2=h,d^2=g,e^2=f,g^2=i,h^2=i,b^a=bf,c^a=cg,d^a=dg,e^a=ef,g^a=gi,h^a=hi,c^b=cg,d^b=dh,g^b=gi,h^b=hi,d^c=di74 conjugacy classes — view the full character table.
| maximal subgroup of | #72257048, #72257090, #72257181, … |
| maximal normal subgroup of | #72257048, #72257090, #72257181, … |
| maximal quotient of | #72264481 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7511387);