Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6573671 |
| 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 |
| prime radical | #63006 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C4×D4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
3004 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 24 points:
(9,10,16,18,15,20,14,19)(11,21,24,12,17,23,22,13);(1,2,4,5)(3,6,7,8)(9,11,15,17)(10,12,20,13)(14,24,16,22)(18,21,19,23);(2,5)(3,7)(9,12)(10,11)(13,15)(14,23)(16,21)(17,20)(18,22)(19,24);(1,3)(2,6)(4,7)(5,8)(9,13,15,12)(10,17,20,11)(14,23,16,21)(18,22,19,24);(2,5)(3,7)(9,14,15,16)(10,18,20,19)(11,22,17,24)(12,23,13,21)Polycyclic presentation:
2^9|a^2=fi,b^2=h,d^2=i,e^2=i,f^2=i,b^a=bf,c^a=cg,d^a=di,e^a=ei,g^a=gi,c^b=ch,e^b=eh,f^b=fi,g^b=gi,d^c=dh,f^c=fi,e^d=eh74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6503377);