Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #486155 |
| 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 | C2×C4⋊C4 | C24 |
| prime radical | #24227 | C2 |
| LCS | G ▷ C2×C4⋊C4 ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ C2×C4×Q8 ◁ G | |
| LECS | G ▷ C2×C4⋊C4 ▷ C23 ▷ C2 ▷ C1 | |
805 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 40 points:
(2,6)(3,4)(7,8)(9,10)(11,15)(12,17)(13,16)(14,19)(18,27)(20,29)(21,28)(22,31)(23,33)(24,35)(25,32)(26,30)(34,38)(36,39)(37,40);(1,2)(3,6)(4,7)(5,8)(9,11,14,22)(10,15,19,31)(12,23,24,20)(13,26,27,21)(16,28,18,30)(17,33,35,29)(25,36,38,37)(32,40,34,39);(1,3,5,4)(2,6,8,7)(10,16)(12,24)(13,27)(15,28)(17,34)(18,19)(20,23)(21,26)(29,40)(30,31)(32,35)(33,39);(1,3,5,4)(2,7,8,6)(9,12,14,24)(10,17,19,35)(11,20,22,23)(13,25,27,38)(15,29,31,33)(16,32,18,34)(21,36,26,37)(28,39,30,40)Polycyclic presentation:
2^9|b^2=i,c^2=g,d^2=gi,e^2=g,f^2=i,b^a=be,c^a=cf,d^a=dg,e^a=eg,f^a=fi,h^a=hi,d^b=dgi,e^b=eg,f^b=fhi,d^c=di,e^c=eh,f^c=fi,f^e=fi47 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 393764);