Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #12019 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C23×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C2×#106 |
| commutator | C2×C8 | C23×C4 |
| Frattini, cosocle | C22×C8 | C24 |
| socle | C22 | C22×#106 |
| prime radical | C24 | C2×SD16 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C22×C42 ◁ C22×C4×C8 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
563 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)(11,19,20,31)(12,15)(13,25)(14,27)(16,21,28,33)(17,26)(18,24)(22,36,34,32)(23,37)(29,38)(30,40,39,35);(2,5)(6,8)(10,15)(16,28)(17,29)(18,27)(19,31)(25,37)(30,39)(32,36);(9,11,13,21,14,22,26,35,12,20,23,33,24,34,38,40)(10,16,17,19,18,30,25,32,15,28,29,31,27,39,37,36);(1,3,4,7)(2,6,5,8)Polycyclic presentation:
2^9|a^2=g,c^2=f,d^2=ei,f^2=h,h^2=i,b^a=be,c^a=cf,f^a=fh,h^a=hi,g^c=gi104 conjugacy classes — view the full character table.
| maximal subgroup of | #72285685, #72285908, #72285918, … |
| maximal normal subgroup of | #72285685, #72285908, #72285918, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 401389);