Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6365322 |
| 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 | |
1530 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 48 points:
(1,2)(3,13)(4,14)(5,9)(6,11)(7,10)(8,12)(15,16)(17,18)(19,27)(20,23)(21,25)(22,33)(24,30)(26,39)(28,45)(29,43)(31,37)(32,36)(34,48)(35,44)(38,42)(40,47)(41,46);(1,3)(2,4)(5,12)(6,9)(7,14)(8,15)(10,16)(11,13)(17,19,23,29)(18,24,20,31)(21,34,36,47)(22,37,39,30)(25,41,32,45)(26,43,33,27)(28,38,46,35)(40,44,48,42);(1,4,8,5)(2,3,12,15)(6,13,7,16)(9,10,14,11)(17,20,23,18)(19,24,29,31)(21,32,36,25)(22,33,39,26)(27,37,43,30)(28,40,46,48)(34,41,47,45)(35,44,38,42);(1,5,8,4)(2,9,12,14)(3,10,15,11)(6,13,7,16)(17,21,23,36)(18,25,20,32)(19,28,29,46)(22,38,39,35)(24,40,31,48)(26,44,33,42)(27,45,43,41)(30,47,37,34);(1,6,8,7)(2,10,12,11)(3,9,15,14)(4,16,5,13)(18,20)(19,29)(25,32)(26,33)(28,46)(30,37)(34,47)(42,44)Polycyclic presentation:
2^9|b^2=hi,c^2=h,d^2=h,e^2=i,f^2=h,g^2=i,b^a=bf,c^a=cg,e^a=eh,f^a=fh,g^a=gi,c^b=ch,d^b=df,e^b=ehi,f^b=fh,e^c=eh,g^c=gi,f^d=fh71 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6272705);