Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6573635 |
| 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 | |
2593 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)(11,17)(12,28)(13,19)(14,16)(15,21)(18,31)(20,34)(22,36)(23,35)(24,26)(25,37)(27,32)(29,40)(30,33)(38,39);(1,3,5,4)(2,6,7,8)(9,11)(10,17)(12,29)(13,23)(14,32)(15,25)(16,27)(18,36)(19,35)(20,24)(21,37)(22,31)(26,34)(28,40)(30,38)(33,39);(9,12,16,19)(10,18,14,30)(11,22,27,39)(13,21,28,20)(15,33,34,31)(17,23,32,40)(24,38,37,36)(25,29,26,35);(1,4,5,3)(2,6,7,8)(9,13,16,28)(10,19,14,12)(11,23,27,40)(15,18,34,30)(17,35,32,29)(20,33,21,31)(22,24,39,37)(25,36,26,38);(9,14)(10,16)(11,24)(12,30)(13,31)(15,21)(17,26)(18,19)(20,34)(22,38)(23,35)(25,32)(27,37)(28,33)(29,40)(36,39)Polycyclic presentation:
2^9|b^2=f,c^2=i,d^2=fi,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,g^c=gi,h^c=hi,e^d=ehi,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, 6484148);