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 | |
1522 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 48 points:
(1,2,6,10)(3,7,12,16)(4,9,5,14)(8,13,15,11)(17,18,23,20)(19,27,29,43)(21,25,36,32)(22,33,39,26)(24,37,31,30)(28,45,46,41)(34,48,47,40)(35,44,38,42);(1,3)(2,7)(4,13)(5,11)(6,12)(8,14)(9,15)(10,16)(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);(1,3)(2,8)(4,11)(5,13)(6,12)(7,14)(9,16)(10,15)(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);(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);(1,4,6,5)(2,9,10,14)(3,11,12,13)(7,8,16,15)(18,20)(19,29)(25,32)(26,33)(28,46)(30,37)(34,47)(42,44)Polycyclic presentation:
2^9|a^2=hi,b^2=h,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, 6272677);