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 | |
3102 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)(11,19,21,22)(12,20,23,24)(13,17)(14,16)(15,18);(1,2)(3,6)(4,5)(7,8)(9,11)(10,12)(13,20)(14,24)(15,21)(16,22)(17,19)(18,23);(2,5)(6,8)(9,12)(10,11)(13,22)(14,19)(15,23)(16,20)(17,24)(18,21);(1,3)(2,6)(4,7)(5,8)(9,13,15,14)(10,16,18,17)(11,20,21,24)(12,22,23,19);(3,7)(6,8)(9,13,15,14)(10,16,18,17)(11,20,21,24)(12,22,23,19)Polycyclic presentation:
2^9|a^2=fg,d^2=i,e^2=i,f^2=i,g^2=i,b^a=bf,c^a=cg,d^a=di,e^a=ei,c^b=ch,f^b=fi,g^c=gi,e^d=eh68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6449423);