Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #491728 |
| 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 | D4×D8 | C22 |
| LCS | G ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C42 ◁ C8×SD16 ◁ G | |
| LECS | G ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
463 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)(9,10,12,17)(11,20,21,15)(13,16,24,31)(14,27,25,19)(18,33,32,26)(22,34,37,28)(23,30,38,36)(29,39,35,40);(1,2,4,7)(3,6,8,5)(9,11,12,21)(10,15,17,20)(13,22,24,37)(14,23,25,38)(16,28,31,34)(18,29,32,35)(19,30,27,36)(26,39,33,40);(10,16)(13,26)(14,25)(15,28)(17,31)(18,19)(20,34)(22,39)(23,38)(24,33)(27,32)(29,30)(35,36)(37,40);(2,6)(3,8)(5,7)Polycyclic presentation:
2^9|a^2=i,b^2=gi,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,f^c=fh,h^c=hi,e^d=eg56 conjugacy classes — view the full character table.
| maximal subgroup of | #72309937 |
| maximal normal subgroup of | #72309937 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 399016);