Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6682638 |
| 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 | |
1791 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 40 points:
(1,2,5,7)(3,8,4,6)(9,10,14,18)(11,21,23,17)(12,24,26,32)(13,19,27,33)(15,29,16,20)(22,37,38,31)(25,40,39,34)(28,36,30,35);(1,2)(3,6)(4,8)(5,7)(9,11,14,23)(10,17,18,21)(12,16,26,15)(13,25,27,39)(19,34,33,40)(20,24,29,32)(22,28,38,30)(31,36,37,35);(9,12,15,23,14,26,16,11)(10,17,20,24,18,21,29,32)(13,25,28,38,27,39,30,22)(19,31,35,40,33,37,36,34);(1,3,5,4)(2,6,7,8)(10,18)(13,27)(17,21)(20,29)(22,38)(24,32)(25,39)(28,30);(1,4,5,3)(2,6,7,8)(9,13)(10,19)(11,22)(12,25)(14,27)(15,28)(16,30)(17,31)(18,33)(20,35)(21,37)(23,38)(24,40)(26,39)(29,36)(32,34)Polycyclic presentation:
2^9|a^2=f,b^2=i,c^2=g,d^2=fi,e^2=fi,g^2=i,h^2=i,b^a=bf,c^a=cg,d^a=df,g^a=gi,c^b=ch,e^b=efh,h^b=hi,h^c=hi,e^d=ei,h^e=hi62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6594185);