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 | |
1971 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)(11,23,12,27)(13,19)(14,21)(15,20)(16,22)(17,26,18,25)(24,35,28,40)(29,34,39,33)(30,37)(31,36)(32,38);(1,2)(3,7)(4,6)(5,8)(9,11)(10,17)(12,16)(13,29)(14,26)(15,25)(18,22)(19,35)(20,27)(21,23)(24,31)(28,30)(32,39)(33,36)(34,37)(38,40);(1,3)(2,5)(4,8)(6,7)(9,12,16,11)(10,18,22,17)(13,24,32,28)(14,26,15,25)(19,33,38,34)(20,27,21,23)(29,30,39,31)(35,37,40,36);(3,8)(5,7)(9,13,15,31,16,32,14,30)(10,19,20,36,22,38,21,37)(11,24,26,29,12,28,25,39)(17,33,23,35,18,34,27,40);(2,6)(5,7)(9,14,16,15)(10,20,22,21)(11,25,12,26)(13,30,32,31)(17,23,18,27)(19,36,38,37)(24,39,28,29)(33,35,34,40)Polycyclic presentation:
2^9|a^2=fg,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, 6914392);