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 | |
1947 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)(11,22,26,36)(12,27,31,39)(13,32)(14,20)(15,21)(16,24,37,25)(17,30,40,29)(18,34)(19,33)(23,38)(28,35);(1,2)(3,7)(4,6)(5,8)(9,11)(10,16)(12,28)(13,29)(14,25)(15,26)(17,38)(18,39)(19,22)(20,36)(21,37)(23,31)(24,33)(27,32)(30,34)(35,40);(1,3)(2,5)(4,8)(6,7)(9,12)(10,17)(11,23)(13,24)(14,29)(15,31)(16,35)(18,22)(19,27)(20,39)(21,40)(25,34)(26,28)(30,33)(32,36)(37,38);(3,8)(5,7)(9,13,33,23,15,34,14,28)(10,18,19,35,21,32,20,38)(11,12,25,30,26,31,24,29)(16,17,36,27,37,40,22,39);(2,6)(5,7)(9,14,15,33)(10,19,21,20)(11,24,26,25)(12,29,31,30)(13,28,34,23)(16,36,37,22)(17,27,40,39)(18,35,32,38)Polycyclic presentation:
2^9|a^2=fgi,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, 6914430);