Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6348447 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C29 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C22×#120 |
| commutator, Frattini, cosocle | C42 | C25 |
| prime radical | #60639 | C2 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C42 ▷ C22 ▷ C1 | |
3042 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 40 points:
(2,5)(3,7)(6,8)(9,10)(11,20)(12,17)(13,18)(14,27)(15,19)(16,22)(21,35)(23,30)(24,32)(25,33)(26,34)(28,31)(29,37)(36,39)(38,40);(3,7)(5,8)(9,11)(10,16)(12,21)(13,22)(14,28)(15,23)(17,29)(18,30)(19,31)(20,27)(24,36)(25,38)(26,37)(32,35)(33,39)(34,40);(1,2)(3,8)(4,6)(5,7)(9,12,15,25)(10,17,19,33)(11,21,23,38)(13,26,14,24)(16,29,31,39)(18,34,27,32)(20,35,30,40)(22,37,28,36);(9,13,15,14)(10,18,19,27)(11,22,23,28)(12,24,25,26)(16,30,31,20)(17,32,33,34)(21,36,38,37)(29,35,39,40);(10,19)(16,31)(17,33)(18,27)(20,30)(29,39)(32,34)(35,40)Polycyclic presentation:
2^9|c^2=h,d^2=h,f^2=h,g^2=i,b^a=bf,c^a=cg,e^a=eh,f^a=fh,g^a=gi,f^b=fh,g^b=gi,d^c=dh,f^c=fi,g^c=gi68 conjugacy classes — view the full character table.
| maximal subgroup of | #72306598, #72306601, #72306604, … |
| maximal normal subgroup of | #72306598, #72306601, #72306604, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6255647);