Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7514148 |
| 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 | |
1738 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 5 generators on 40 points:
(1,2)(3,8)(4,6)(5,7)(9,10)(11,23)(12,26)(13,17)(14,22)(15,21)(16,20)(18,29)(19,31)(24,39)(25,34)(27,38)(28,35)(30,37)(32,36)(33,40);(1,3,5,4)(2,6,7,8)(9,11,15,13)(10,17,21,23)(12,27,30,40)(14,25,16,31)(18,35,36,39)(19,20,34,22)(24,32,28,29)(26,33,37,38);(1,4,5,3)(2,6,7,8)(9,12,15,30)(10,18,21,36)(11,24,13,28)(14,29,16,32)(17,33,23,38)(19,35,34,39)(20,26,22,37)(25,40,31,27);(9,13,15,11)(10,19,21,34)(12,28,30,24)(14,31,16,25)(17,22,23,20)(18,35,36,39)(26,33,37,38)(27,32,40,29);(2,7)(6,8)(9,14,15,16)(10,20,21,22)(11,25,13,31)(12,29,30,32)(17,34,23,19)(18,26,36,37)(24,40,28,27)(33,39,38,35)Polycyclic presentation:
2^9|b^2=f,c^2=f,d^2=i,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,g^c=gi,h^c=hi,g^d=gi74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7422637);