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 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1798 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 40 points:
(3,7)(6,8)(9,10,15,21,16,22,14,20)(11,23,25,17,26,28,18,12)(13,19,32,37,33,38,31,36)(24,39,35,30,40,34,29,27);(2,5)(3,7)(9,11,15,25,16,26,14,18)(10,17,20,23,22,12,21,28)(13,29,31,40,33,35,32,24)(19,34,37,27,38,39,36,30);(1,2)(3,6)(4,5)(7,8)(9,12)(10,18)(11,20)(13,30)(14,23)(15,28)(16,17)(19,35)(21,26)(22,25)(24,37)(27,33)(29,38)(31,34)(32,39)(36,40);(2,5)(6,8)(9,13,16,33)(10,19,22,38)(11,24,26,40)(12,27,17,30)(14,31,15,32)(18,29,25,35)(20,36,21,37)(23,39,28,34);(1,3,4,7)(2,6,5,8)(9,14,16,15)(10,20,22,21)(11,25,26,18)(12,23,17,28)(13,31,33,32)(19,36,38,37)(24,35,40,29)(27,39,30,34)Polycyclic presentation:
2^9|a^2=g,b^2=f,d^2=i,e^2=h,f^2=i,g^2=i,b^a=bf,c^a=cg,e^a=ehi,f^a=fi,c^b=ch,d^b=df,e^b=eh,d^c=dh,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, 7056605);