Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of D4×#453 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C29 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C22×#120 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C4×D4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
3219 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 5 generators on 24 points:
(9,10,15,17)(11,18,21,22)(12,20,23,19)(13,16,14,24);(1,2,4,5)(3,6,7,8)(9,11)(10,12)(13,19)(14,20)(15,21)(16,22)(17,23)(18,24);(2,5)(3,7)(9,12)(10,11)(13,22)(14,18)(15,23)(16,19)(17,21)(20,24);(1,3)(2,6)(4,7)(5,8)(9,13,15,14)(10,16,17,24)(11,19,21,20)(12,22,23,18);(2,5)(3,7)Polycyclic presentation:
2^9|a^2=i,b^2=h,d^2=i,f^2=i,g^2=i,b^a=bf,c^a=cg,f^a=fi,g^a=gi,c^b=ch,e^b=eh,f^b=fi,d^c=dh,g^c=gi,e^d=eh80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6502856);