Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #11651 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C2×#102 |
| commutator | C2×C8 | C23×C4 |
| Frattini, cosocle | C22×C8 | C24 |
| socle | C22 | C22×#102 |
| prime radical | C2×#466 | C22 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C22×C42 ◁ C22×C4×C8 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
867 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 40 points:
(1,2)(3,6)(4,5)(7,8)(9,10,14,19,28,32,30,18,15,20,27,38,12,17,13,29)(11,21,24,16,22,34,23,33,26,39,40,36,25,37,31,35);(9,11,12,22,15,26,28,25)(10,16,32,35,20,36,17,33)(13,24,27,23,30,40,14,31)(18,37,38,39,29,34,19,21);(2,5)(6,8)(9,12,15,28)(10,17,20,32)(11,22,26,25)(13,27,30,14)(16,33,36,35)(18,19,29,38)(21,34,39,37)(23,40,31,24);(1,3,4,7)(2,6,5,8)Polycyclic presentation:
2^9|a^2=g,b^2=egh,c^2=i,d^2=f,e^2=h,g^2=hi,h^2=i,b^a=be,c^a=cf,e^a=ei,e^b=eh,g^b=gh,h^b=hi92 conjugacy classes — view the full character table.
| maximal subgroup of | #72310187, #72310200, #72310220, … |
| maximal normal subgroup of | #72310187, #72310200, #72310220, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 402728);