Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6996405 |
| 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 | |
1950 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)(11,23,26,37)(12,27,31,40)(13,32)(14,21)(15,20)(16,22)(17,25,38,29)(18,30,36,34)(19,33)(24,35)(28,39);(1,2)(3,7)(4,6)(5,8)(9,11)(10,17)(12,28)(13,30)(14,29)(15,25)(16,26)(18,39)(19,40)(20,23)(21,37)(22,38)(24,31)(27,32)(33,34)(35,36);(1,3)(2,5)(4,8)(6,7)(9,12)(10,18)(11,24)(13,25)(14,30)(15,34)(16,31)(17,35)(19,37)(20,40)(21,27)(22,36)(23,32)(26,28)(29,33)(38,39);(3,8)(5,7)(9,13,15,24,16,33,14,28)(10,19,20,39,22,32,21,35)(11,12,29,34,26,31,25,30)(17,36,37,27,38,18,23,40);(2,6)(5,7)(9,14,16,15)(10,20,22,21)(11,25,26,29)(12,30,31,34)(13,28,33,24)(17,37,38,23)(18,40,36,27)(19,39,32,35)Polycyclic presentation:
2^9|a^2=fg,d^2=fi,e^2=i,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=gi62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6914422);