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 |
| prime radical | #24279 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1883 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)(25,40)(26,29)(27,36)(28,34)(31,38)(35,39);(1,3,5,4)(2,6,8,7)(9,11)(10,15)(12,25)(13,23)(14,21)(16,33)(17,35)(18,20)(19,32)(22,29)(24,30)(26,31)(27,39)(28,38)(34,37)(36,40);(10,16)(11,21)(14,30)(15,20)(17,34)(18,19)(22,25)(23,24)(26,36)(28,29)(31,37)(38,39);(1,3,5,4)(2,7,8,6)(9,10)(11,15)(12,26)(13,18)(14,19)(16,30)(17,29)(20,23)(21,32)(22,35)(24,33)(25,31)(27,34)(28,36)(37,39)(38,40);(1,4,5,3)(2,6,8,7)(9,12)(10,17)(11,22)(13,27)(14,28)(15,31)(16,34)(18,36)(19,26)(20,37)(21,25)(23,38)(24,39)(29,30)(32,40)(33,35)Polycyclic presentation:
2^9|b^2=fi,d^2=fi,e^2=fi,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, 6485852);