Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6573635 |
| 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 |
| prime radical | #60584 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1875 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)(4,6)(5,8)(9,10)(11,18)(12,28)(13,19)(14,20)(15,22)(16,25)(17,24)(21,32)(23,40)(26,35)(27,39)(29,38)(30,37)(31,36)(33,34);(1,3,5,4)(2,6,8,7)(9,11,14,25)(10,16,20,18)(12,13,30,29)(15,26,32,27)(17,36,34,40)(19,28,38,37)(21,35,22,39)(23,33,31,24);(1,3,5,4)(2,7,8,6)(9,12,14,30)(10,17,20,34)(11,23,25,31)(13,27,29,26)(15,33,32,24)(16,19,18,38)(21,37,22,28)(35,40,39,36);(1,3,5,4)(2,7,8,6)(9,11)(10,18)(12,29)(13,30)(14,25)(15,26)(16,20)(17,36)(19,37)(21,39)(22,35)(23,33)(24,31)(27,32)(28,38)(34,40);(1,4,5,3)(2,6,8,7)(9,13)(10,19)(11,24)(12,27)(14,29)(15,23)(16,34)(17,18)(20,38)(21,36)(22,40)(25,33)(26,30)(28,39)(31,32)(35,37)Polycyclic presentation:
2^9|b^2=f,c^2=f,d^2=fi,e^2=fi,g^2=i,h^2=i,b^a=bf,c^a=cg,g^a=gi,c^b=ch,d^b=di,e^b=eh,h^b=hi,d^c=dh,g^c=gi,h^c=hi,e^d=eh,h^d=hi,h^e=hi74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6486236);