Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6573652 |
| 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 | |
2619 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)(11,22)(12,28)(13,19)(14,20)(15,32)(16,21)(17,26)(18,30)(23,38)(24,29)(25,27)(31,34)(33,37)(35,36)(39,40);(1,3,5,4)(2,6,7,8)(9,11,14,26)(10,17,20,22)(12,29,19,36)(13,24,28,35)(15,27,16,33)(18,38,34,40)(21,25,32,37)(23,30,39,31);(9,12)(10,18)(11,23)(13,21)(14,19)(15,31)(16,30)(17,35)(20,34)(22,24)(25,38)(26,39)(27,36)(28,32)(29,33)(37,40);(1,4,5,3)(2,6,7,8)(9,13,14,28)(10,19,20,12)(11,24,26,35)(15,18,16,34)(17,36,22,29)(21,31,32,30)(23,37,39,25)(27,38,33,40);(9,10)(11,25)(12,18)(13,31)(14,20)(15,21)(16,32)(17,33)(19,34)(22,27)(23,38)(24,36)(26,37)(28,30)(29,35)(39,40)Polycyclic presentation:
2^9|b^2=f,d^2=f,g^2=i,h^2=i,b^a=bf,c^a=cg,g^a=gi,c^b=ch,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, 6483926);