Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6993875 |
| 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 | |
2077 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)(9,10,15,21)(11,22,17,27)(12,25,16,24)(13,18,31,38)(14,19,32,20)(23,39,34,35)(26,40,33,28)(29,36,30,37);(1,2)(3,7)(4,6)(5,8)(9,11,15,17)(10,16,21,12)(13,28,31,40)(14,24,32,25)(18,35,38,39)(19,22,20,27)(23,29,34,30)(26,36,33,37);(1,3)(2,5)(4,8)(6,7)(9,12,15,16)(10,17,21,11)(13,26,31,33)(14,27,32,22)(18,34,38,23)(19,25,20,24)(28,36,40,37)(29,35,30,39);(3,8)(5,7)(9,13)(10,18)(11,23)(12,26)(14,29)(15,31)(16,33)(17,34)(19,36)(20,37)(21,38)(22,39)(24,28)(25,40)(27,35)(30,32);(2,6)(5,7)(9,14,15,32)(10,19,21,20)(11,24,17,25)(12,27,16,22)(13,29,31,30)(18,36,38,37)(23,28,34,40)(26,35,33,39)Polycyclic presentation:
2^9|a^2=i,b^2=i,c^2=i,e^2=i,f^2=i,g^2=i,b^a=bf,c^a=cg,e^a=eh,f^a=fi,g^a=gi,c^b=ch,d^b=df,e^b=eh,f^b=fi,d^c=dh,g^c=gi,f^d=fi62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6901082);