Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #61629 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C4 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C23×D4 |
| commutator | C2×C4 | C24×C4 |
| Frattini, cosocle | C22×C4 | C25 |
| socle | C22 | C22×C4×D4 |
| prime radical | #60749 | C2 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C2×#408 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1948 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 5 generators on 40 points:
(1,2)(3,7)(4,6)(5,8)(9,10,15,20,16,21,14,31)(11,17,23,12,24,26,18,27)(13,19,30,35,28,34,29,40)(22,32,37,25,36,39,33,38);(1,3)(2,5)(4,8)(6,7)(9,11,15,23,16,24,14,18)(10,17,20,12,21,26,31,27)(13,22,30,37,28,36,29,33)(19,32,35,25,34,39,40,38);(9,12,15,26,16,27,14,17)(10,18,20,11,21,23,31,24)(13,25,30,39,28,38,29,32)(19,33,35,22,34,37,40,36);(3,8)(5,7)(9,13)(10,19)(11,22)(12,25)(14,29)(15,30)(16,28)(17,32)(18,33)(20,35)(21,34)(23,37)(24,36)(26,39)(27,38)(31,40);(2,6)(5,7)(13,28)(19,34)(22,36)(25,38)(29,30)(32,39)(33,37)(35,40)Polycyclic presentation:
2^9|a^2=h,b^2=h,c^2=h,g^2=i,h^2=i,b^a=bf,c^a=cg,e^a=ef,g^a=gi,d^b=df,g^c=gi,e^d=ei104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6301874);