Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6574077 |
| 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 | |
1960 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)(9,10,16,22)(11,17,26,37)(12,19,29,38)(13,20,31,24)(14,32,15,21)(18,34,39,28)(23,35,40,30)(25,36,33,27);(1,3,5,4)(2,6,7,8)(9,11)(10,17)(12,24)(13,19)(14,25)(15,33)(16,26)(18,35)(20,29)(21,27)(22,37)(23,28)(30,39)(31,38)(32,36)(34,40);(9,12,15,34,16,29,14,28)(10,18,32,38,22,39,21,19)(11,23,33,24,26,40,25,20)(13,27,35,17,31,36,30,37);(1,4,5,3)(2,6,7,8)(9,12)(10,19)(11,24)(13,17)(14,28)(15,34)(16,29)(18,32)(20,26)(21,39)(22,38)(23,25)(27,30)(31,37)(33,40)(35,36);(9,13)(10,20)(11,18)(12,27)(14,30)(15,35)(16,31)(17,34)(19,25)(21,40)(22,24)(23,32)(26,39)(28,37)(29,36)(33,38)Polycyclic presentation:
2^9|a^2=i,b^2=f,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=ehi,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, 6491537);