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 | #52360 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
3905 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 48 points:
(1,2,7,11)(3,8,6,16)(4,5,10,9)(12,14,13,15)(17,18,22,21)(19,23,28,26)(20,24,29,30)(25,31,27,32)(33,34,39,37)(35,42,41,38)(36,44,46,40)(43,45,47,48);(1,3,7,6)(2,8,11,16)(4,13,10,12)(5,15,9,14)(17,19)(18,23)(20,27)(21,26)(22,28)(24,32)(25,29)(30,31)(34,37)(35,41)(40,44)(43,47);(1,4,7,10)(2,9,11,5)(3,12,6,13)(8,15,16,14)(17,20,22,29)(18,24,21,30)(19,25,28,27)(23,31,26,32)(33,35)(34,38)(36,43)(37,42)(39,41)(40,45)(44,48)(46,47);(1,4)(2,5)(3,13)(6,12)(7,10)(8,15)(9,11)(14,16)(17,21,22,18)(19,26,28,23)(20,24,29,30)(25,31,27,32)(33,36)(34,40)(35,43)(37,44)(38,45)(39,46)(41,47)(42,48);(1,5)(2,10)(3,14)(4,11)(6,15)(7,9)(8,13)(12,16)(17,20)(18,24)(19,27)(21,30)(22,29)(23,32)(25,28)(26,31)(33,37,39,34)(35,42,41,38)(36,44,46,40)(43,48,47,45);(1,6,7,3)(2,8,11,16)(4,12,10,13)(5,15,9,14)(17,19)(18,23)(20,27)(21,26)(22,28)(24,32)(25,29)(30,31)(33,38,39,42)(34,41,37,35)(36,45,46,48)(40,47,44,43)Polycyclic presentation:
2^9|a^2=i,b^2=gh,c^2=gi,d^2=hi,e^2=g,f^2=h,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dgi,e^c=egi,f^c=fgi,e^d=egi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8900854);