Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #3415 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C8 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C22×D8 |
| commutator | C8 | C23×C8 |
| Frattini, cosocle | C4×C8 | C24 |
| socle | C22 | C2×C4×D8 |
| prime radical | C22×D16 | C4 |
| LCS | G ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ #431 ◁ #1450 ◁ G | |
| LECS | G ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
538 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 40 points:
(1,2,4,7,5,3,6,8)(9,10)(11,16)(12,15)(13,26)(14,17)(18,23)(19,24)(20,32)(21,27)(22,36)(25,28)(29,37)(30,35)(31,34)(33,39)(38,40);(10,15)(12,23)(13,24)(16,27)(17,28)(18,19)(21,35)(22,34)(25,37)(26,29)(30,31)(32,39)(33,40)(36,38);(1,3,4,8,5,2,6,7)(9,11)(10,16)(12,21)(13,22)(14,20)(15,27)(17,32)(18,30)(19,31)(23,35)(24,34)(25,33)(26,36)(28,39)(29,38)(37,40);(10,17)(11,20)(15,28)(18,29)(19,26)(21,33)(22,34)(35,40)Polycyclic presentation:
2^9|a^2=f,c^2=f,e^2=gi,f^2=h,g^2=i,b^a=be,d^a=di,e^a=eg,g^a=gi,e^b=eg,g^b=gi,d^c=di128 conjugacy classes — view the full character table.
| maximal subgroup of | #72232006, #72232009, #72232010, … |
| maximal normal subgroup of | #72232006, #72232009, #72232010, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 390700);