Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6682658 |
| 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 | |
1891 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)(3,7)(4,6)(5,8)(9,10,15,20)(11,22,25,16)(12,27,31,34)(13,21,28,18)(14,19,36,40)(17,32,38,29)(23,35,39,33)(24,30,26,37);(1,3,4,8)(2,5,6,7)(9,11,15,25)(10,16,20,22)(12,19,31,40)(13,26,28,24)(14,34,36,27)(17,35,38,33)(18,37,21,30)(23,29,39,32);(9,12,28,32,15,31,13,29)(10,17,21,34,20,38,18,27)(11,23,24,40,25,39,26,19)(14,30,35,22,36,37,33,16);(2,6)(5,7)(9,13,15,28)(10,18,20,21)(11,24,25,26)(12,29,31,32)(14,35,36,33)(16,30,22,37)(17,27,38,34)(19,23,40,39);(3,8)(5,7)(9,14)(10,19)(11,17)(12,30)(13,33)(15,36)(16,29)(18,39)(20,40)(21,23)(22,32)(24,34)(25,38)(26,27)(28,35)(31,37)Polycyclic presentation:
2^9|a^2=i,b^2=f,c^2=gi,d^2=i,g^2=i,h^2=i,b^a=bf,c^a=cg,d^a=df,g^a=gi,c^b=ch,d^b=di,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, 6590479);