Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1468 |
| automorphism | ? |
| homology | Abelianization (H1) = C43 · Schur multiplier (H2) = C26×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C22×C4 | C2×C22⋊C4 |
| commutator | C23 | C43 |
| Frattini, cosocle | C24×C4 | C23 |
| socle | C23 | C4×C22⋊C4 |
| prime radical | C25 | C22×C4 |
| LCS | G ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ C24×C8 ◁ G | |
| LECS | G ▷ C24×C4 ▷ C2 ▷ C1 | |
1790 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 3 generators on 40 points:
(1,2)(3,4)(5,6)(7,8)(9,10)(11,15,20,26)(12,16)(13,18)(14,17)(19,25,31,35)(21,27,33,37)(22,29)(23,28)(24,30)(32,36,39,40)(34,38);(2,4)(9,11)(10,15,17,26)(12,19)(13,21)(14,20)(16,25,28,35)(18,27,30,37)(22,32)(23,31)(24,33)(29,36,38,40)(34,39);(6,8)(9,12,13,22,14,23,24,34)(10,16,18,29,17,28,30,38)(11,19,21,32,20,31,33,39)(15,25,27,36,26,35,37,40)Polycyclic presentation:
2^9|a^2=f,b^2=g,c^2=h,h^2=i,b^a=bd,c^a=ce,d^a=di,g^a=gi,d^b=di,f^b=fi128 conjugacy classes — view the full character table.
| maximal subgroup of | #72314510, #72314517, #72314526, … |
| maximal normal subgroup of | #72314510, #72314517, #72314526, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 30652);