Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6993899 |
| 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 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1691 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 40 points:
(3,7)(6,8)(9,10,15,20,16,21,14,19)(11,22,24,17,12,25,27,33)(13,18,30,37,31,38,29,36)(23,39,40,32,26,34,28,35);(2,5)(3,7)(9,11)(10,17)(12,16)(13,28)(14,27)(15,24)(18,34)(19,22)(20,25)(21,33)(23,30)(26,29)(31,40)(32,36)(35,37)(38,39);(1,2)(3,6)(4,5)(7,8)(9,12,16,11)(10,17,21,33)(13,23,31,26)(14,27,15,24)(18,35,38,32)(19,25,20,22)(28,29,40,30)(34,36,39,37);(2,5)(6,8)(9,13)(10,18)(11,23)(12,26)(14,29)(15,30)(16,31)(17,32)(19,36)(20,37)(21,38)(22,39)(24,40)(25,34)(27,28)(33,35);(1,3,4,7)(2,6,5,8)(9,14,16,15)(10,19,21,20)(11,24,12,27)(13,29,31,30)(17,25,33,22)(18,36,38,37)(23,40,26,28)(32,34,35,39)Polycyclic presentation:
2^9|a^2=g,c^2=i,e^2=h,f^2=i,g^2=i,b^a=bf,c^a=cg,e^a=ehi,f^a=fi,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, 7055703);