Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #180873 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C42 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator | C2×C4 | C22×C42 |
| Frattini, cosocle | C22×C8 | C24 |
| socle | C22 | C22×#97 |
| prime radical | C22×D4 | M16 |
| LCS | G ▷ C2×C4 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
643 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 40 points:
(1,2)(3,7)(4,6)(5,8)(9,10,13,19,14,16,27,29,15,20,26,34,12,18,25,33)(11,17,22,31,24,28,36,38,23,32,37,40,21,30,35,39);(1,3)(2,5)(4,8)(6,7)(10,16,20,18)(11,21,23,24)(13,26)(19,33,34,29)(22,36,37,35)(25,27)(31,40)(38,39);(3,8)(5,7)(9,11,13,22,14,24,27,36,15,23,26,37,12,21,25,35)(10,17,19,31,16,28,29,38,20,32,34,40,18,30,33,39);(2,6)(5,7)Polycyclic presentation:
2^9|a^2=f,b^2=g,c^2=f,e^2=i,f^2=h,h^2=i,b^a=be,d^a=deh,g^a=gi,c^b=ce,f^b=fi,g^c=gi110 conjugacy classes — view the full character table.
| maximal subgroup of | #72239337, #72239463, #72239465, … |
| maximal normal subgroup of | #72239337, #72239463, #72239465, … |
| maximal quotient of | #72343334, #72343378, #72343410, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 400516);