Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6365694 |
| 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 | C42 | C25 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C42 ▷ C22 ▷ C1 | |
1553 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 48 points:
(1,2)(3,8)(4,13)(5,11)(6,15)(7,10)(9,14)(12,16)(17,18)(19,27)(20,24)(21,34)(22,25)(23,31)(26,35)(28,41)(29,38)(30,37)(32,45)(33,39)(36,44)(40,46)(42,48)(43,47);(1,3,7,4)(2,8,10,13)(5,14,6,12)(9,15,16,11)(17,19)(18,23)(20,28)(21,29)(22,30)(24,36)(25,38)(26,37)(27,34)(31,35)(32,42)(33,43)(39,48)(40,47)(41,45)(44,46);(1,4,7,3)(2,9,10,16)(5,14,6,12)(8,15,13,11)(17,20,22,33)(18,24,25,39)(19,28,30,43)(21,32,35,46)(23,36,38,48)(26,40,34,45)(27,41,37,47)(29,42,31,44);(2,10)(8,13)(9,16)(11,15)(19,29)(21,35)(23,37)(26,34)(27,38)(28,42)(30,31)(32,46)(36,47)(40,45)(41,48)(43,44);(1,5,7,6)(2,11,10,15)(3,12,4,14)(8,16,13,9)(18,25)(19,30)(24,39)(26,34)(28,43)(29,31)(40,45)(42,44)Polycyclic presentation:
2^9|b^2=i,c^2=hi,e^2=i,f^2=h,g^2=i,b^a=bf,c^a=cg,d^a=di,e^a=eh,f^a=fh,g^a=gi,d^b=df,e^b=ehi,f^b=fh,g^c=gi,f^d=fh59 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6273342);