Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6667964 |
| 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 | |
1737 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 40 points:
(1,2,5,7)(3,8,4,6)(9,10)(11,17)(12,27)(13,19)(14,20)(15,21)(16,22)(18,30)(23,40)(24,34)(25,35)(26,37)(28,33)(29,38)(31,39)(32,36);(1,2)(3,6)(4,8)(5,7)(9,11)(10,17)(12,28)(13,24)(14,30)(15,32)(16,26)(18,20)(19,34)(21,36)(22,37)(23,29)(25,31)(27,33)(35,39)(38,40);(9,12,16,25)(10,18,22,38)(11,23,26,14)(13,29,15,30)(17,33,37,39)(19,27,21,35)(20,34,40,36)(24,31,32,28);(1,3,5,4)(2,6,7,8)(9,13,16,15)(10,19,22,21)(11,24,26,32)(12,29,25,30)(14,28,23,31)(17,34,37,36)(18,27,38,35)(20,33,40,39);(1,4,5,3)(2,6,7,8)(9,14,16,23)(10,20,22,40)(11,25,26,12)(13,31,15,28)(17,35,37,27)(18,34,38,36)(19,39,21,33)(24,29,32,30)Polycyclic presentation:
2^9|a^2=f,c^2=i,d^2=fi,e^2=fi,g^2=i,h^2=i,b^a=bf,c^a=cg,d^a=df,g^a=gi,c^b=ch,e^b=efh,h^b=hi,g^c=gi,h^c=hi,e^d=ei,h^e=hi62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6583719);