Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6996395 |
| 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 | |
1974 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)(11,21,12,25)(13,18)(14,31)(15,19)(16,20)(17,24,33,23)(22,34,26,39)(27,35,38,32)(28,40)(29,36)(30,37);(1,2)(3,7)(4,6)(5,8)(9,11)(10,17)(12,16)(13,27)(14,24)(15,23)(18,34)(19,25)(20,33)(21,31)(22,29)(26,28)(30,38)(32,36)(35,40)(37,39);(1,3)(2,5)(4,8)(6,7)(9,12,16,11)(10,17,20,33)(13,22,30,26)(14,24,15,23)(18,35,37,32)(19,21,31,25)(27,28,38,29)(34,36,39,40);(3,8)(5,7)(9,13,15,29,16,30,14,28)(10,18,19,36,20,37,31,40)(11,22,24,27,12,26,23,38)(17,32,21,34,33,35,25,39);(2,6)(5,7)(9,14,16,15)(10,19,20,31)(11,23,12,24)(13,28,30,29)(17,21,33,25)(18,36,37,40)(22,38,26,27)(32,34,35,39)Polycyclic presentation:
2^9|a^2=fgi,c^2=i,d^2=f,e^2=i,f^2=i,g^2=i,b^a=bf,c^a=cg,e^a=eh,c^b=ch,d^b=df,e^b=eh,f^b=fi,d^c=dh,g^c=gi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6914400);