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 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2624 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)(11,16)(12,27)(13,18)(14,20)(15,21)(17,23)(19,32)(22,35)(24,40)(25,33)(26,34)(28,39)(29,37)(30,36)(31,38);(1,3,5,4)(2,6,7,8)(9,11)(10,16)(12,28)(13,29)(14,24)(15,26)(17,35)(18,37)(19,33)(20,40)(21,34)(22,23)(25,32)(27,39)(30,31)(36,38);(9,12)(10,17)(11,22)(13,25)(14,31)(15,29)(16,18)(19,37)(20,27)(21,38)(23,32)(24,28)(26,30)(33,36)(34,39)(35,40);(1,4,5,3)(2,6,7,8)(9,11)(10,16)(12,13)(14,24)(15,26)(17,36)(18,27)(19,33)(20,40)(21,34)(22,31)(23,30)(25,32)(28,29)(35,38)(37,39);(9,13)(10,18)(11,23)(12,25)(14,30)(15,28)(16,17)(19,35)(20,36)(21,39)(22,32)(24,29)(26,31)(27,33)(34,38)(37,40)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=ehi,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, 6484142);