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 | |
2111 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)(9,10,14,19,15,21,31,20)(11,12,24,26,17,27,23,16)(13,18,29,34,30,36,37,35)(22,25,28,39,33,40,38,32);(1,2)(3,7)(4,6)(5,8)(9,11,15,17)(10,16,21,26)(12,19,27,20)(13,28,30,38)(14,23,31,24)(18,25,36,40)(22,37,33,29)(32,35,39,34);(1,3)(2,5)(4,8)(6,7)(9,12,14,26,15,27,31,16)(10,17,20,24,21,11,19,23)(13,25,29,39,30,40,37,32)(18,33,35,28,36,22,34,38);(3,8)(5,7)(9,13,14,29,15,30,31,37)(10,18,19,34,21,36,20,35)(11,22,24,28,17,33,23,38)(12,25,26,39,27,40,16,32);(2,6)(5,7)(9,14,15,31)(10,19,21,20)(11,23,17,24)(12,16,27,26)(13,29,30,37)(18,34,36,35)(22,38,33,28)(25,32,40,39)Polycyclic presentation:
2^9|a^2=f,b^2=i,c^2=g,d^2=f,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, 6928580);