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 | #3489 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1893 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)(11,20)(12,17)(13,18)(14,16)(15,23)(19,30)(21,32)(22,37)(24,33)(25,38)(26,28)(27,35)(29,36)(31,39)(34,40);(1,3,5,4)(2,6,8,7)(9,11)(10,15)(12,25)(13,23)(14,21)(16,33)(17,34)(18,20)(19,32)(22,29)(24,30)(26,37)(27,40)(28,39)(31,36)(35,38);(1,3,5,4)(2,7,8,6)(10,16)(11,21)(14,30)(15,20)(17,26)(18,19)(22,25)(23,24)(28,29)(31,37)(35,36)(39,40);(1,4,5,3)(2,6,8,7)(9,10)(11,15)(12,26)(13,18)(14,19)(16,30)(17,28)(20,23)(21,32)(22,38)(24,33)(25,37)(27,36)(29,35)(31,40)(34,39);(9,12,13,27)(10,17,18,35)(11,22,23,39)(14,28,30,29)(15,31,20,37)(16,26,19,36)(21,25,24,40)(32,38,33,34)Polycyclic presentation:
2^9|b^2=fi,c^2=fi,d^2=fi,e^2=i,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, 6484805);