Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #12087 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C23×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #442 |
| commutator | C2×C8 | C23×C4 |
| Frattini, cosocle | C22×C8 | C24 |
| socle | C22 | C2×#442 |
| prime radical | C2×#135 | C23 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C2×C42 ◁ C4×#97 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
524 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)(11,19,21,33)(12,15)(13,25)(14,27)(16,22,28,37)(17,26)(18,24)(20,34,35,32)(23,39)(29,40)(30,36,31,38);(2,5)(6,8)(10,15)(11,20,21,35)(13,23)(16,30,28,31)(18,27)(19,32,33,34)(22,38,37,36)(26,40);(9,11,13,22,14,20,26,36,12,21,23,37,24,35,40,38)(10,16,17,19,18,31,25,32,15,28,29,33,27,30,39,34);(1,3,4,7)(2,6,5,8)Polycyclic presentation:
2^9|a^2=g,b^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,c^b=ch,f^b=fi,g^c=gi68 conjugacy classes — view the full character table.
| maximal subgroup of | #72310187, #72310191, #72310194, … |
| maximal normal subgroup of | #72310187, #72310191, #72310194, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 402516);