Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6533075 |
| 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 | |
3089 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 5 generators on 24 points:
(9,10,13,22,14,16,24,17)(11,18,23,12,19,21,20,15);(1,2)(3,6)(4,5)(7,8)(9,11)(10,15)(12,16)(13,23)(14,19)(17,21)(18,22)(20,24);(2,5)(6,8)(10,16)(11,19)(13,24)(18,21);(1,3)(2,6)(4,7)(5,8)(9,12,14,15)(10,11,16,19)(13,21,24,18)(17,20,22,23);(3,7)(6,8)(10,16)(12,15)(17,22)(18,21)Polycyclic presentation:
2^9|a^2=fg,d^2=i,f^2=i,b^a=bf,c^a=cg,d^a=di,e^a=ei,f^a=fi,g^a=gi,c^b=ch,f^b=fi,g^b=gi,f^c=fi,e^d=eh68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6449626);