Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6574063 |
| 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 | |
1987 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)(3,8)(4,6)(5,7)(9,10)(11,17)(12,19)(13,20)(14,31)(15,21)(16,22)(18,26)(23,33)(24,34)(25,36)(27,35)(28,37)(29,39)(30,40)(32,38);(1,3,5,4)(2,6,7,8)(9,11,16,12)(10,17,22,19)(13,27,30,28)(14,24,15,26)(18,31,34,21)(20,35,40,37)(23,32,25,29)(33,38,36,39);(9,12,15,24,16,11,14,26)(10,18,31,17,22,34,21,19)(13,25,32,28,30,23,29,27)(20,35,39,33,40,37,38,36);(1,4,5,3)(2,6,7,8)(9,12)(10,19)(11,16)(13,28)(14,26)(15,24)(17,22)(18,31)(20,37)(21,34)(23,32)(25,29)(27,30)(33,38)(35,40)(36,39);(9,13)(10,20)(11,23)(12,25)(14,29)(15,32)(16,30)(17,33)(18,35)(19,36)(21,38)(22,40)(24,28)(26,27)(31,39)(34,37)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,d^b=di,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, 6491520);