Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6996799 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×D42 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1973 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)(11,20,24,31)(12,25)(13,18)(14,19)(15,22,16,23)(17,28)(21,36,37,33)(26,30,38,32)(27,34)(29,39)(35,40);(1,2)(3,7)(4,6)(5,8)(9,11)(10,15)(12,26)(13,23)(14,24)(16,19)(17,33)(18,31)(20,34)(21,29)(22,27)(25,36)(28,38)(30,35)(32,39)(37,40);(1,3)(2,5)(4,8)(6,7)(9,11)(10,16)(12,21)(13,22)(14,24)(15,19)(17,32)(18,31)(20,34)(23,27)(25,30)(26,29)(28,37)(33,39)(35,36)(38,40);(3,8)(5,7)(9,12,27,40,14,28,13,29)(10,17,34,39,19,25,18,35)(11,21,23,38,24,37,22,26)(15,30,31,36,16,32,20,33);(2,6)(5,7)(12,28)(17,25)(21,37)(26,38)(29,40)(30,32)(33,36)(35,39)Polycyclic presentation:
2^9|a^2=fgi,d^2=fi,f^2=i,g^2=i,b^a=bf,c^a=cg,d^a=di,e^a=eh,c^b=ch,d^b=df,e^b=eh,f^b=fi,d^c=dh,g^c=gi,e^d=ei62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6914445);