Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #33522 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C22×D8 |
| commutator | C2×C8 | C23×C4 |
| Frattini, cosocle | C4×C8 | C24 |
| socle | C22 | C2×C4×D8 |
| prime radical | C22×D8 | C2×C4 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ #431 ◁ #1450 ◁ G | |
| LECS | G ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
476 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 40 points:
(1,2,4,3,5,6,8,7)(9,10,13,17)(11,16,20,31)(12,15,25,28)(14,26,24,19)(18,33,29,23)(21,27,34,38)(22,37,35,32)(30,40,39,36);(10,15)(12,23)(14,24)(16,27)(17,28)(18,19)(21,36)(22,35)(25,33)(26,29)(30,32)(31,38)(34,40)(37,39);(1,3,4,6,5,7,8,2)(9,11)(10,16)(12,21)(13,20)(14,22)(15,27)(17,31)(18,30)(19,32)(23,36)(24,35)(25,34)(26,37)(28,38)(29,39)(33,40);(10,17)(11,20)(15,28)(18,29)(19,26)(21,34)(22,35)(36,40)Polycyclic presentation:
2^9|a^2=f,c^2=fi,e^2=gi,f^2=h,g^2=i,b^a=be,c^a=ch,d^a=di,e^a=eg,g^a=gi,e^b=eg,g^b=gi,d^c=di92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 390990);