Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7519075 |
| 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)(11,23)(12,28)(13,31)(14,22)(15,21)(16,20)(17,27)(18,19)(24,39)(25,29)(26,36)(30,32)(33,38)(34,40)(35,37);(2,6)(7,8)(9,11)(10,17)(12,29)(13,25)(14,26)(15,27)(16,33)(18,37)(19,24)(20,36)(21,23)(22,38)(28,34)(30,35)(31,40)(32,39);(1,3,5,4)(2,7,6,8)(9,12,14,30,15,31,16,19)(10,18,20,13,21,32,22,28)(11,24,33,40,27,35,26,29)(17,34,38,39,23,25,36,37);(1,4,5,3)(2,8,6,7)(9,13,16,18,15,28,14,32)(10,19,22,31,21,30,20,12)(11,25,26,39,27,34,33,37)(17,35,36,40,23,24,38,29);(2,6)(7,8)(9,14,15,16)(10,20,21,22)(11,26,27,33)(12,30,31,19)(13,32,28,18)(17,36,23,38)(24,29,35,40)(25,39,34,37)Polycyclic presentation:
2^9|c^2=fg,d^2=fgi,e^2=i,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,d^c=di,h^d=hi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7468998);