Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #488895 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×#447 |
| commutator, Frattini, cosocle | C4×C8 | C24 |
| prime radical | #61409 | C2 |
| LCS | G ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ #2960 ◁ G | |
| LECS | G ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
817 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 40 points:
(2,5)(3,7)(6,8)(9,10,14,19)(11,16,20,30)(12,15,25,29)(13,26,24,18)(17,33,28,23)(21,27,34,38)(22,37,35,32)(31,40,39,36);(3,7)(5,8)(10,15)(12,23)(13,24)(16,27)(17,18)(19,29)(21,36)(22,35)(25,33)(26,28)(30,38)(31,32)(34,40)(37,39);(1,2,4,6)(3,8,7,5)(9,11)(10,16)(12,21)(13,22)(14,20)(15,27)(17,31)(18,32)(19,30)(23,36)(24,35)(25,34)(26,37)(28,39)(29,38)(33,40);(11,20)(16,30)(21,34)(22,35)(27,38)(31,39)(32,37)(36,40)Polycyclic presentation:
2^9|a^2=i,c^2=h,e^2=gi,f^2=h,g^2=i,b^a=be,c^a=cf,e^a=eg,f^a=fh,g^a=gi,e^b=eg,f^b=fh,g^b=gi,d^c=di,e^c=eh,f^c=fh68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 396373);