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 | |
1981 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)(10,15,19,18)(11,20,23,22)(16,29)(17,24,32,31)(21,34,37,36)(28,33)(30,39)(35,40);(1,2)(3,7)(4,6)(5,8)(9,10)(11,16)(12,24)(13,18)(14,19)(15,25)(17,26)(20,28)(21,35)(22,33)(23,29)(27,31)(30,36)(32,38)(34,39)(37,40);(1,3)(2,5)(4,8)(6,7)(9,11,14,23)(10,16,19,29)(12,21,27,37)(13,22,25,20)(15,28,18,33)(17,30,32,39)(24,35,31,40)(26,36,38,34);(3,8)(5,7)(9,12,25,38,14,27,13,26)(10,17,18,31,19,32,15,24)(11,21,20,34,23,37,22,36)(16,30,33,40,29,39,28,35);(2,6)(5,7)Polycyclic presentation:
2^9|a^2=fgi,c^2=i,d^2=fi,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, 6914401);