Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #508066 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×B(4,𝔽2) |
| commutator, Frattini, cosocle | C22×D4 | C24 |
| prime radical | C2×#120 | C23 |
| LCS | G ▷ C22×D4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ C2×C4×D4 ◁ G | |
| LECS | G ▷ C22×D4 ▷ C22 ▷ C2 ▷ C1 | |
817 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 40 points:
(2,3)(6,7)(9,10)(11,19)(12,23)(13,17)(14,18)(15,21)(16,25)(20,34)(22,32)(24,38)(26,31)(27,29)(28,40)(30,39)(33,36)(35,37);(1,2)(3,5)(4,6)(7,8)(9,11,13,21)(10,15,17,19)(12,20,24,35)(14,27,26,22)(16,28,30,33)(18,32,31,29)(23,37,38,34)(25,36,39,40);(1,3,5,2)(4,6,8,7)(9,12,13,24)(10,16,17,30)(11,20,21,35)(14,25,26,39)(15,28,19,33)(18,23,31,38)(22,36,27,40)(29,37,32,34);(1,4,5,8)(2,6,3,7)(10,17)(12,24)(15,19)(18,31)(20,35)(25,39)(29,32)(36,40)Polycyclic presentation:
2^9|b^2=i,c^2=e,d^2=ei,b^a=be,c^a=cf,d^a=di,g^a=ghi,c^b=cg,f^b=fh,d^c=de,h^c=hi,g^f=gi41 conjugacy classes — view the full character table.
| maximal subgroup of | #72317541, #72317587, #72317612, … |
| maximal normal subgroup of | #72317541, #72317587, #72317612, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 415459);