Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7518936 |
| 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 | C24 | C22×D4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C4×C22⋊Q8 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1882 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 28 points:
(5,6)(7,11)(8,12)(9,10)(13,14,20,22)(15,21,26,23)(16,19,17,18)(24,28,25,27);(1,2)(3,4)(6,10)(11,12)(13,15,20,26)(14,21,22,23)(16,27,17,28)(18,25,19,24);(5,7,9,8)(6,11,10,12)(13,16,18,14,20,17,19,22)(15,23,24,28,26,21,25,27);(2,4)(5,7,9,8)(6,11,10,12)(13,17,18,22,20,16,19,14)(15,21,24,27,26,23,25,28);(5,8,9,7)(6,11,10,12)Polycyclic presentation:
2^9|a^2=i,b^2=i,c^2=fg,d^2=fg,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=cgi,d^b=dh,g^b=gi,h^b=hi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7472297);