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 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4129 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 48 points:
(1,2,6,8)(3,9,4,7)(5,13,14,10)(11,15,12,16)(17,18,23,21)(19,22,27,24)(20,28,29,25)(26,31,32,30)(33,34)(35,40)(36,38)(37,44)(39,42)(41,45)(43,46)(47,48);(1,3,6,4)(2,7,8,9)(5,11,14,12)(10,15,13,16)(17,19,23,27)(18,24,21,22)(20,26,29,32)(25,31,28,30)(35,42)(39,40)(43,48)(46,47);(2,8)(7,9)(10,13)(15,16)(20,29)(25,28)(26,32)(30,31)(33,35,38,42)(34,39,36,40)(37,43,45,48)(41,46,44,47);(1,2)(3,9)(4,7)(5,10)(6,8)(11,16)(12,15)(13,14)(17,20)(18,25)(19,26)(21,28)(22,30)(23,29)(24,31)(27,32)(33,35)(34,40)(36,39)(37,43)(38,42)(41,47)(44,46)(45,48);(1,4,6,3)(2,9,8,7)(5,12,14,11)(10,16,13,15)(17,21,23,18)(19,22,27,24)(20,25,29,28)(26,31,32,30)(33,36,38,34)(35,40,42,39)(37,44,45,41)(43,47,48,46);(1,5,6,14)(2,10,8,13)(3,11,4,12)(7,15,9,16)(17,22)(18,19)(20,30)(21,27)(23,24)(25,26)(28,32)(29,31)(33,37,38,45)(34,41,36,44)(35,43,42,48)(39,46,40,47)Polycyclic presentation:
2^9|a^2=g,b^2=g,c^2=i,e^2=gi,f^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^d=egi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8072756);