Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6500002 |
| 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 | C22×C4 | C25 |
| prime radical | #62821 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
3178 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 5 generators on 40 points:
(9,10)(11,20)(12,24,26,33)(13,27)(14,19)(15,18)(16,23)(17,25,34,29)(21,36,39,40)(22,31)(28,32)(30,37,35,38);(1,2)(3,7)(4,6)(5,8)(9,11)(10,16)(12,21)(13,22)(14,23)(15,28)(17,30)(18,31)(19,20)(24,36)(25,37)(26,39)(27,32)(29,38)(33,40)(34,35);(1,3)(2,5)(4,8)(6,7)(9,12)(10,17)(11,21)(13,25)(14,26)(15,29)(16,30)(18,33)(19,34)(20,35)(22,37)(23,39)(24,27)(28,38)(31,40)(32,36);(3,8)(5,7)(9,13,14,15)(10,18,19,27)(11,22,23,28)(12,25,26,29)(16,31,20,32)(17,33,34,24)(21,37,39,38)(30,40,35,36);(2,6)(5,7)Polycyclic presentation:
2^9|a^2=fi,d^2=i,g^2=i,b^a=bf,c^a=cg,d^a=di,c^b=ch,e^b=eh,g^b=gi,d^c=dh,f^c=fi,g^c=gi68 conjugacy classes — view the full character table.
| maximal subgroup of | #72319333 |
| maximal normal subgroup of | #72319333 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6407201);