Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7500736 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×D42 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | #60644 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2332 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)(9,10,16,11)(12,23,28,24)(13,29,32,34)(14,22,15,19)(17,26,20,27)(18,30,21,31)(25,33,39,35)(36,40,37,38);(1,3,5,2)(4,7,8,6)(9,11,16,10)(12,24,28,23)(13,21,32,18)(14,22,15,19)(17,26,20,27)(25,37,39,36)(29,31,34,30)(33,38,35,40);(1,4,5,8)(2,6,3,7)(9,12)(10,17)(11,20)(13,25)(14,26)(15,27)(16,28)(18,33)(19,24)(21,35)(22,23)(29,36)(30,38)(31,40)(32,39)(34,37);(2,3)(6,7)(9,13)(10,18)(11,21)(12,25)(14,30)(15,31)(16,32)(17,33)(19,34)(20,35)(22,29)(23,36)(24,37)(26,38)(27,40)(28,39);(9,14,16,15)(10,19,11,22)(12,26,28,27)(13,30,32,31)(17,24,20,23)(18,34,21,29)(25,38,39,40)(33,37,35,36)Polycyclic presentation:
2^9|a^2=i,b^2=fi,c^2=f,e^2=i,g^2=i,h^2=i,b^a=bf,c^a=cg,d^a=dfg,e^a=ei,g^a=gi,c^b=ch,d^b=df,h^b=hi,g^c=gi,h^c=hi,g^d=gi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7409021);