Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6574084 |
| 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 |
| prime radical | #60461 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1981 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)(3,8)(4,6)(5,7)(9,10)(11,21)(12,17)(13,18)(14,20)(15,25)(16,30)(19,28)(22,34)(23,37)(24,27)(26,33)(29,36)(31,35)(32,39)(38,40);(1,3,5,4)(2,6,7,8)(9,11)(10,15)(12,23)(13,25)(14,26)(16,34)(17,32)(18,21)(19,33)(20,24)(22,31)(27,28)(29,39)(30,38)(35,40)(36,37);(9,12,28,31,13,29,14,30)(10,16,20,36,18,35,19,17)(11,22,27,39,25,38,26,23)(15,32,24,34,21,37,33,40);(1,4,5,3)(2,6,7,8)(9,12,13,29)(10,17,18,36)(11,23,25,39)(14,30,28,31)(15,32,21,37)(16,19,35,20)(22,26,38,27)(24,34,33,40);(9,10)(11,24)(12,16)(13,18)(14,19)(15,26)(17,30)(20,28)(21,27)(22,34)(23,32)(25,33)(29,35)(31,36)(37,39)(38,40)Polycyclic presentation:
2^9|b^2=fi,c^2=gi,d^2=f,g^2=i,h^2=i,b^a=bf,c^a=cg,g^a=gi,c^b=ch,e^b=eh,h^b=hi,d^c=dh,h^c=hi,e^d=eh,h^d=hi,h^e=hi74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6491514);