Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #489733 |
| 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 |
| LCS | G ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ #2965 ◁ G | |
| LECS | G ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
760 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)(11,19)(12,23)(13,28)(14,16)(15,21)(17,27)(18,26)(20,35)(22,37)(24,34)(25,38)(29,33)(30,36)(31,39)(32,40);(1,2)(3,6)(4,8)(5,7)(9,11)(10,15)(12,24)(13,29)(14,20)(16,30)(17,33)(18,34)(19,23)(21,27)(22,26)(25,39)(28,31)(32,37)(35,38)(36,40);(1,2)(3,6)(4,8)(5,7)(12,25)(17,32)(19,35)(21,36)(23,38)(24,39)(27,40)(33,37);(1,3,5,4)(2,6,7,8)(10,16)(11,20)(17,32)(18,28)(21,36)(22,29)(23,38)(24,39)Polycyclic presentation:
2^9|d^2=hi,e^2=gi,f^2=hi,g^2=i,b^a=be,c^a=cf,d^a=dh,e^a=eg,f^a=fhi,g^a=gi,d^b=di,e^b=eg,f^b=fh,g^b=gi,e^c=eh,f^c=fhi47 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 397191);