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 | C2×#121 | C23 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
3878 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 48 points:
(1,2,5,8)(3,6,10,15)(4,11,13,7)(9,16,12,14)(21,29)(23,30)(26,32)(28,31)(34,38)(36,43)(39,44)(42,47);(1,2)(3,6)(4,7)(5,8)(9,14)(10,15)(11,13)(12,16)(17,18,22,24)(19,25,27,20)(21,23,29,30)(26,31,32,28)(35,41)(39,44)(42,47)(46,48);(1,3,5,10)(2,6,8,15)(4,12,13,9)(7,16,11,14)(17,19,22,27)(18,20,24,25)(21,26,29,32)(23,28,30,31)(33,34)(35,39)(36,40)(37,38)(41,44)(42,46)(43,45)(47,48);(1,4,5,13)(2,7,8,11)(3,9,10,12)(6,14,15,16)(17,20)(18,19)(21,28)(22,25)(23,26)(24,27)(29,31)(30,32)(33,35)(34,39)(36,42)(37,41)(38,44)(40,46)(43,47)(45,48);(1,3)(2,6)(4,9)(5,10)(7,14)(8,15)(11,16)(12,13)(17,20)(18,19)(21,28)(22,25)(23,26)(24,27)(29,31)(30,32)(33,36,37,43)(34,40,38,45)(35,42,41,47)(39,46,44,48);(1,2)(3,6)(4,7)(5,8)(9,14)(10,15)(11,13)(12,16)(17,21)(18,23)(19,26)(20,28)(22,29)(24,30)(25,31)(27,32)Polycyclic presentation:
2^9|a^2=g,b^2=i,c^2=gi,d^2=g,e^2=h,b^a=bg,c^a=ch,d^a=dg,e^a=eh,f^a=fgi,c^b=ci,d^b=dh,d^c=dgi,e^c=egi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8034575);