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 | |
2093 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)(12,13)(14,22)(15,21)(16,20)(17,27)(18,30)(19,33)(24,39)(25,28)(26,36)(29,32)(31,40)(34,38)(35,37);(2,6)(7,8)(9,11,15,27)(10,17,21,23)(12,28,29,40)(13,31,32,25)(14,26,16,34)(18,37,33,39)(19,35,30,24)(20,36,22,38);(1,3,5,4)(2,7,6,8)(9,12,16,30,15,29,14,19)(10,18,22,13,21,33,20,32)(11,24,26,28,27,35,34,40)(17,31,36,37,23,25,38,39);(1,4,5,3)(2,8,6,7)(9,13,16,33,15,32,14,18)(10,19,22,12,21,30,20,29)(11,25,26,39,27,31,34,37)(17,35,36,40,23,24,38,28);(2,6)(7,8)(9,14,15,16)(10,20,21,22)(11,26,27,34)(12,19,29,30)(13,18,32,33)(17,36,23,38)(24,28,35,40)(25,39,31,37)Polycyclic presentation:
2^9|b^2=i,c^2=fgi,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,h^d=hi74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7468978);