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 | |
4126 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)(3,11,12,7)(4,13,14,9)(5,10,15,16)(17,18)(19,24)(20,25)(21,30)(22,23)(26,32)(27,31)(28,29)(34,36)(39,43)(40,44)(47,48);(1,3,6,12)(2,7,8,11)(4,5,14,15)(9,10,13,16)(17,18)(19,24)(20,28)(21,26)(22,23)(25,29)(27,31)(30,32)(33,34,38,36)(35,39,42,43)(37,40,46,44)(41,47,45,48);(2,8)(4,14)(5,15)(7,11)(18,23)(19,27)(20,29)(21,32)(33,35,38,42)(34,39,36,43)(37,45,46,41)(40,48,44,47);(1,4,6,14)(2,9,8,13)(3,5,12,15)(7,10,11,16)(17,19,22,27)(18,24,23,31)(20,30,29,26)(21,28,32,25)(33,36,38,34)(35,39,42,43)(37,44,46,40)(41,47,45,48);(1,5)(2,10)(3,4)(6,15)(7,9)(8,16)(11,13)(12,14)(17,20)(18,25)(19,26)(21,31)(22,29)(23,28)(24,32)(27,30)(33,35)(34,39)(36,43)(37,41)(38,42)(40,47)(44,48)(45,46);(1,2)(3,7)(4,9)(5,10)(6,8)(11,12)(13,14)(15,16)(17,21,22,32)(18,26,23,30)(19,28,27,25)(20,31,29,24)(33,37,38,46)(34,40,36,44)(35,41,42,45)(39,47,43,48)Polycyclic presentation:
2^9|a^2=hi,b^2=g,c^2=ghi,d^2=gi,f^2=gh,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, 8900558);