Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #501164 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×B(4,𝔽2) |
| commutator, Frattini, cosocle | C22×D4 | C24 |
| prime radical | C2×#560 | C22 |
| LCS | G ▷ C22×D4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ C2×C4×D4 ◁ G | |
| LECS | G ▷ C22×D4 ▷ C22 ▷ C2 ▷ C1 | |
973 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)(3,7)(4,6)(5,8)(9,10)(11,15)(12,22)(13,18)(14,17)(16,24)(19,32)(20,29)(21,28)(23,35)(25,36)(26,31)(27,33)(30,38)(34,40)(37,39);(1,3,5,4)(2,6,8,7)(9,11,14,21)(10,15,17,28)(12,23,25,19)(13,20,26,34)(16,27,30,39)(18,29,31,40)(22,35,36,32)(24,33,38,37);(1,3,5,4)(2,7,8,6)(9,12,13,24)(10,16,18,22)(11,19,20,33)(14,25,26,38)(15,27,29,32)(17,30,31,36)(21,23,34,37)(28,39,40,35);(1,4,5,3)(2,7,8,6)(10,17)(15,28)(16,30)(18,31)(22,36)(27,39)(29,40)(32,35)Polycyclic presentation:
2^9|b^2=ei,c^2=ef,d^2=e,b^a=be,c^a=cf,d^a=dei,g^a=ghi,c^b=cg,f^b=fh,g^c=gh,h^c=hi,g^f=gi47 conjugacy classes — view the full character table.
| maximal subgroup of | #72317538, #72317548, #72317620, … |
| maximal normal subgroup of | #72317538, #72317548, #72317620, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 410647);