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 | |
1928 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 40 points:
(1,2)(3,5)(4,6)(7,8)(9,10)(11,23,27,37)(12,28,30,39)(13,31)(14,20)(15,21)(16,22)(17,26,38,25)(18,29,36,34)(19,33)(24,35)(32,40);(1,2)(3,7)(4,6)(5,8)(9,11)(10,17)(12,24)(13,29)(14,25)(15,26)(16,27)(18,35)(19,39)(20,37)(21,23)(22,38)(28,31)(30,32)(33,34)(36,40);(1,3)(2,5)(4,8)(6,7)(9,12)(10,18)(11,24)(13,25)(14,29)(15,34)(16,30)(17,35)(19,23)(20,28)(21,39)(22,36)(26,33)(27,32)(31,37)(38,40);(3,8)(5,7)(9,13,15,32,16,33,14,24)(10,19,21,35,22,31,20,40)(11,12,25,34,27,30,26,29)(17,36,37,28,38,18,23,39);(2,6)(5,7)(9,14,16,15)(10,20,22,21)(11,25,27,26)(12,29,30,34)(13,32,33,24)(17,37,38,23)(18,28,36,39)(19,35,31,40)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=gi,e^d=ei62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6914438);