Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| automorphism | ? |
| homology | Abelianization (H1) = C26 · Schur multiplier (H2) = C212 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C23 | C26 |
| prime radical | #392 | C23 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
3785 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 48 points:
(1,2)(3,9)(4,8)(5,13)(6,16)(7,10)(11,14)(12,15)(17,18,21,24)(19,22,25,29)(20,27,28,23)(26,32,31,30)(34,38)(35,41)(36,44)(46,48);(1,3,7,13)(2,5,10,9)(4,11,6,12)(8,15,16,14)(19,25)(22,29)(26,31)(30,32)(33,34,37,38)(35,42,41,39)(36,40,44,45)(43,48,47,46);(1,4,7,6)(2,8,10,16)(3,11,13,12)(5,15,9,14)(17,18)(19,22)(20,23)(21,24)(25,29)(26,30)(27,28)(31,32)(33,35,37,41)(34,39,38,42)(36,43,44,47)(40,46,45,48);(1,5,7,9)(2,3,10,13)(4,14,6,15)(8,12,16,11)(17,19,21,25)(18,22,24,29)(20,26,28,31)(23,30,27,32)(33,34)(35,39)(36,40)(37,38)(41,42)(43,46)(44,45)(47,48);(1,3)(2,9)(4,11)(5,10)(6,12)(7,13)(8,14)(15,16)(17,20,21,28)(18,23,24,27)(19,26,25,31)(22,30,29,32)(33,36,37,44)(34,40,38,45)(35,43,41,47)(39,46,42,48);(1,6,7,4)(2,8,10,16)(3,12,13,11)(5,15,9,14)Polycyclic presentation:
2^9|a^2=hi,b^2=g,c^2=g,d^2=gh,e^2=h,f^2=gi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,f^a=fgi,c^b=ci,d^b=dh,e^b=egi,d^c=dgi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8022694);