Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #509 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C8 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C42 | C2×D8 |
| commutator | C8 | C23×C8 |
| Frattini, cosocle | C4×C8 | C24 |
| socle | C22 | C2×C4×D8 |
| prime radical | C2×#508 | C4 |
| LCS | G ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C42 ◁ C2×C4×C8 ◁ C2×C82 ◁ G | |
| LECS | G ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
514 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 4 generators on 40 points:
(1,2,4,7,5,3,6,8)(9,10)(11,20)(12,18)(13,25)(14,29)(15,17)(16,22)(19,27)(21,37)(23,36)(24,34)(26,32)(28,39)(30,35)(31,38)(33,40);(9,11)(10,16)(12,21)(13,26)(14,30)(15,23)(17,31)(18,32)(19,35)(20,25)(22,27)(24,38)(28,40)(29,33)(34,37)(36,39);(1,3,4,8,5,2,6,7)(10,17)(16,31)(18,29)(19,34)(20,36)(25,39)(32,33)(35,37);(9,12,15,14)(10,18,17,29)(11,21,23,30)(13,24,28,27)(16,32,31,33)(19,25,34,39)(20,37,36,35)(22,26,38,40)Polycyclic presentation:
2^9|a^2=f,c^2=f,d^2=i,e^2=gi,f^2=h,g^2=i,b^a=be,c^a=ci,e^a=eg,g^a=gi,e^b=eg,g^b=gi128 conjugacy classes — view the full character table.
| maximal subgroup of | C8×#2179, #72223673, #72223678, … |
| maximal normal subgroup of | C8×#2179, #72223673, #72223678, … |
| maximal quotient of | C2×#483499, #72286742 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 390695);