Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7021375 |
| 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 | |
2114 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)(3,5)(4,6)(7,8)(9,10,14,20,15,21,30,19)(11,12,23,26,17,27,24,16)(13,18,31,36,32,37,39,35)(22,25,28,40,34,29,38,33);(1,2)(3,7)(4,6)(5,8)(9,11)(10,16)(12,19)(13,28)(14,24)(15,17)(18,25)(20,27)(21,26)(22,31)(23,30)(29,37)(32,38)(33,36)(34,39)(35,40);(1,3)(2,5)(4,8)(6,7)(9,12,14,26,15,27,30,16)(10,17,19,23,21,11,20,24)(13,29,31,33,32,25,39,40)(18,22,35,38,37,34,36,28);(3,8)(5,7)(9,13,30,39,15,32,14,31)(10,18,19,35,21,37,20,36)(11,22,24,38,17,34,23,28)(12,25,16,33,27,29,26,40);(2,6)(5,7)(9,14,15,30)(10,19,21,20)(11,23,17,24)(12,16,27,26)(13,31,32,39)(18,35,37,36)(22,28,34,38)(25,33,29,40)Polycyclic presentation:
2^9|a^2=f,c^2=g,d^2=fi,e^2=i,f^2=i,g^2=i,b^a=bf,c^a=cg,e^a=eh,g^a=gi,c^b=ch,d^b=df,e^b=eh,f^b=fi,d^c=dh,e^c=ei62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6928574);