Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #491760 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | D4×D8 |
| commutator, Frattini, cosocle | C4×C8 | C24 |
| prime radical | #33838 | C2 |
| LCS | G ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C42 ◁ C8×SD16 ◁ G | |
| LECS | G ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
470 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)(3,8)(6,7)(9,10)(11,19)(12,15)(13,25)(14,28)(16,21)(17,27)(18,24)(20,35)(22,37)(23,38)(26,34)(29,33)(30,36)(31,40)(32,39);(1,2,4,7)(3,6,8,5)(10,15)(16,30)(17,31)(18,28)(19,35)(25,38)(32,34)(33,37);(9,11)(10,16)(12,20)(13,26)(14,29)(15,30)(17,33)(18,34)(19,25)(21,27)(22,24)(23,39)(28,32)(31,37)(35,38)(36,40);(2,6)(3,8)(5,7)(11,20)(16,30)(19,35)(21,36)(22,29)(26,39)(32,34)(33,37)Polycyclic presentation:
2^9|b^2=g,e^2=g,f^2=hi,h^2=i,b^a=be,c^a=cf,e^a=eg,f^a=fh,h^a=hi,d^b=dei,e^b=eg,d^c=di,f^c=fh,h^c=hi,e^d=eg50 conjugacy classes — view the full character table.
| maximal subgroup of | #72287679 |
| maximal normal subgroup of | #72287679 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 399020);