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 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2089 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)(11,21)(12,27)(13,18)(14,31)(15,20)(16,24)(17,29)(19,32)(22,36)(23,39)(25,35)(26,34)(28,33)(30,37)(38,40);(1,2)(3,7)(4,6)(5,8)(9,11)(10,16)(12,28)(13,22)(14,25)(15,26)(17,36)(18,38)(19,34)(20,35)(21,31)(23,29)(24,32)(27,39)(30,40)(33,37);(1,3,4,8)(2,5,6,7)(9,12)(10,17)(11,22)(13,15)(14,30)(16,33)(18,19)(20,37)(21,38)(23,25)(24,40)(26,28)(27,31)(29,32)(34,39)(35,36);(3,8)(5,7)(9,13,15,12)(10,18,20,27)(11,23,26,40)(14,30,32,29)(16,33,35,36)(17,31,37,19)(21,39,34,38)(22,24,28,25);(2,6)(5,7)(9,14,15,32)(10,19,20,31)(11,24,26,25)(12,29,13,30)(16,21,35,34)(17,27,37,18)(22,23,28,40)(33,39,36,38)Polycyclic presentation:
2^9|c^2=hi,d^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,e^c=ei,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, 6962126);