Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6667760 |
| 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 | |
1728 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 40 points:
(1,2,4,5)(3,7,8,6)(9,10,14,18)(11,20,24,15)(12,26,29,34)(13,19,31,17)(16,28,35,30)(21,38,39,33)(22,23,25,36)(27,32,40,37);(1,2)(3,6)(4,5)(7,8)(9,11)(10,15)(12,27)(13,22)(14,24)(16,33)(17,23)(18,20)(19,36)(21,30)(25,31)(26,37)(28,39)(29,40)(32,34)(35,38);(2,5)(6,7)(9,12,14,29)(10,16,18,35)(11,21,24,39)(13,28,31,30)(15,32,20,37)(17,34,19,26)(22,27,25,40)(23,38,36,33);(1,3,4,8)(2,6,5,7)(9,13,14,31)(10,17,18,19)(11,22,24,25)(12,28,29,30)(15,23,20,36)(16,34,35,26)(21,27,39,40)(32,38,37,33);(9,10)(11,23)(12,16)(13,17)(14,18)(15,25)(19,31)(20,22)(21,38)(24,36)(26,30)(27,37)(28,34)(29,35)(32,40)(33,39)Polycyclic presentation:
2^9|a^2=f,c^2=i,d^2=f,g^2=i,h^2=i,b^a=bf,c^a=cg,d^a=df,e^a=ei,g^a=gi,c^b=ch,e^b=efh,h^b=hi,g^c=gi,h^c=hi,h^e=hi62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6578200);