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 |
| prime radical | #62955 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C4×D4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
3045 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 24 points:
(9,10,16,18,15,19,14,24)(11,20,22,12,17,23,21,13);(1,2)(3,6)(4,5)(7,8)(9,11,15,17)(10,12,19,13)(14,21,16,22)(18,23,24,20);(2,5)(6,8)(9,12,15,13)(10,17,19,11)(14,23,16,20)(18,22,24,21);(1,3)(2,6)(4,7)(5,8)(9,13,15,12)(10,17,19,11)(14,23,16,20)(18,21,24,22);(3,7)(6,8)(9,14,15,16)(10,18,19,24)(11,21,17,22)(12,23,13,20)Polycyclic presentation:
2^9|a^2=fg,b^2=i,c^2=i,d^2=i,e^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, 6449636);