Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #393989 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C2×#127 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| socle | C22 | C2×#392 |
| prime radical | #60444 | C2 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1740 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)(3,5)(4,6)(7,8)(9,10)(11,15)(12,22)(13,25)(14,18)(16,24)(17,26)(19,32)(20,28)(21,36)(23,38)(27,34)(29,35)(30,40)(31,39)(33,37);(1,3,4,7)(2,5,6,8)(9,11)(10,15)(12,19)(13,21)(14,20)(16,30)(17,29)(18,28)(22,37)(23,33)(24,34)(25,36)(26,35)(27,31)(32,38)(39,40);(9,12,13,24,14,23,26,39)(10,16,17,22,18,31,25,38)(11,19,21,34,20,33,35,40)(15,27,29,32,28,30,36,37);(2,6)(5,8)(11,20)(15,28)(19,33)(21,35)(27,30)(29,36)(32,37)(34,40)Polycyclic presentation:
2^9|b^2=h,c^2=f,f^2=i,b^a=be,c^a=cf,d^a=dh,f^a=fi,g^a=gi,c^b=cg,d^b=di,e^c=ei68 conjugacy classes — view the full character table.
| maximal subgroup of | #72257232, #72257240, #72313120, … |
| maximal normal subgroup of | #72257232, #72257240, #72313120, … |
| maximal quotient of | #72257659, #72257676, #72257759, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 313388);