Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #7624128 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C23×D8 |
| commutator, Frattini, cosocle | C2×C8 | C25 |
| prime radical | C2×#3287 | C2 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C4×D4 ◁ C2×C8×D4 ◁ G | |
| LECS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
1552 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)(11,21,22,38)(12,19,14,17)(13,26,29,34)(16,31,32,24)(18,25,36,28)(23,39,30,35)(27,37,40,33);(1,3)(2,5)(4,8)(6,7)(11,22)(16,32)(21,38)(23,30)(24,31)(27,40)(33,37)(35,39);(9,11,15,22)(10,16,20,32)(12,23,14,30)(13,27,29,40)(17,33,19,37)(18,35,36,39)(21,34,38,26)(24,25,31,28);(3,8)(5,7)(9,12,15,14)(10,17,20,19)(11,23,22,30)(13,25,29,28)(16,33,32,37)(18,34,36,26)(21,35,38,39)(24,27,31,40);(2,6)(5,7)Polycyclic presentation:
2^9|a^2=i,c^2=i,d^2=i,g^2=hi,h^2=i,b^a=bf,c^a=cg,d^a=di,e^a=ef,g^a=gh,h^a=hi,c^b=ci,d^b=df,g^c=gh,h^c=hi71 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7531343);