Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #32941 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C2×#121 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| socle | C22 | C22×#121 |
| prime radical | C2×#3050 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2224 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 5 generators on 40 points:
(1,2)(3,5)(4,6)(7,8)(9,10)(11,20)(12,25)(13,19)(14,29)(15,17)(16,23)(18,28)(21,35)(22,30)(24,33)(26,40)(27,32)(31,36)(34,38)(37,39);(1,3,4,7)(2,5,6,8)(9,11,13,22,15,24,14,23)(10,16,29,33,17,30,19,20)(12,21,26,36,28,38,27,37)(18,31,40,35,25,39,32,34);(2,6)(5,8)(10,17)(16,30)(18,25)(19,29)(20,33)(31,39)(32,40)(34,35);(9,12,14,27,15,28,13,26)(10,18,19,32,17,25,29,40)(11,21,23,37,24,38,22,36)(16,31,20,34,30,39,33,35);(9,13,15,14)(10,19,17,29)(11,22,24,23)(12,26,28,27)(16,20,30,33)(18,32,25,40)(21,36,38,37)(31,34,39,35)Polycyclic presentation:
2^9|b^2=fg,d^2=f,e^2=i,f^2=i,b^a=bf,c^a=cg,d^a=dh,f^a=fi,h^d=hi104 conjugacy classes — view the full character table.
| maximal subgroup of | #72306936, #72306950, #72306951, … |
| maximal normal subgroup of | #72306936, #72306950, #72306951, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6536900);