Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #501172 |
| 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×#127 | C23 |
| LCS | G ▷ C22×D4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ C2×C4×D4 ◁ G | |
| LECS | G ▷ C22×D4 ▷ C22 ▷ C2 ▷ C1 | |
937 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)(9,10,14,18)(11,15,20,28)(12,22,25,37)(13,17,26,32)(16,30,31,24)(19,33,23,36)(21,29,35,40)(27,38,39,34);(1,3,5,4)(2,6,8,7)(9,11)(10,15)(12,23)(13,21)(14,20)(16,27)(17,29)(18,28)(19,25)(22,36)(24,34)(26,35)(30,38)(31,39)(32,40)(33,37);(1,3,5,4)(2,7,8,6)(9,12,13,24)(10,16,17,22)(11,19,21,34)(14,25,26,30)(15,27,29,33)(18,31,32,37)(20,23,35,38)(28,39,40,36);(1,4,5,3)(2,7,8,6)(11,20)(15,28)(19,23)(21,35)(27,39)(29,40)(33,36)(34,38)Polycyclic presentation:
2^9|a^2=i,b^2=e,c^2=ef,d^2=e,b^a=be,c^a=cf,d^a=de,g^a=ghi,c^b=cg,d^b=di,f^b=fh,g^c=gh,h^c=hi,g^f=gi47 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 410684);