Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7519063 |
| 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 | |
2087 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)(3,7)(4,8)(5,6)(9,10)(11,16)(12,23)(13,26)(14,30)(15,19)(17,28)(18,24)(20,35)(21,34)(22,32)(25,38)(27,39)(29,40)(31,36)(33,37);(2,6)(7,8)(9,11,15,22)(10,16,19,32)(12,24,21,14)(13,27,29,20)(17,31,33,38)(18,34,30,23)(25,28,36,37)(26,39,40,35);(1,3,5,4)(2,7,6,8)(9,12,14,11,15,21,24,22)(10,16,18,23,19,32,30,34)(13,25,28,20,29,36,37,27)(17,31,26,35,33,38,40,39);(1,4,5,3)(2,8,6,7)(9,13,24,37,15,29,14,28)(10,17,30,40,19,33,18,26)(11,20,12,25,22,27,21,36)(16,31,34,39,32,38,23,35);(2,6)(7,8)Polycyclic presentation:
2^9|b^2=i,c^2=fg,d^2=fgi,g^2=i,b^a=bf,c^a=cg,d^a=dg,e^a=ef,g^a=gi,h^a=hi,c^b=cg,d^b=dh,g^b=gi,h^d=hi74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7468973);