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 | |
2074 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)(11,20)(12,23)(13,26)(14,18)(15,30)(16,22)(17,29)(19,27)(21,35)(24,32)(25,34)(28,39)(31,38)(33,40)(36,37);(2,6)(7,8)(9,11,14,22)(10,16,18,20)(12,15,25,27)(13,21,28,36)(17,32,33,31)(19,34,30,23)(24,29,38,40)(26,37,39,35);(1,3,5,4)(2,7,6,8)(9,12,15,22,14,25,27,11)(10,16,19,23,18,20,30,34)(13,24,29,36,28,38,40,21)(17,31,26,37,33,32,39,35);(1,4,5,3)(2,8,6,7)(9,13,27,40,14,28,15,29)(10,17,30,39,18,33,19,26)(11,21,25,38,22,36,12,24)(16,31,34,35,20,32,23,37);(2,6)(7,8)(10,18)(16,20)(17,33)(19,30)(23,34)(26,39)(31,32)(35,37)Polycyclic presentation:
2^9|b^2=i,c^2=fgi,d^2=fg,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, 7468975);