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 | |
4101 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 48 points:
(2,7)(3,10)(9,14)(11,16)(17,18,22,23)(19,25,27,21)(20,28,29,24)(26,31,32,30)(33,34,39,42)(35,40,38,36)(37,41,47,48)(43,45,46,44);(1,2,6,7)(3,8,10,4)(5,9,13,14)(11,15,16,12)(17,19,22,27)(18,21,23,25)(20,26,29,32)(24,31,28,30)(33,35,39,38)(34,40,42,36)(37,43,47,46)(41,45,48,44);(3,10)(4,8)(5,13)(9,14)(18,23)(20,29)(21,25)(26,32)(33,36,39,40)(34,35,42,38)(37,45,47,44)(41,46,48,43);(1,3,6,10)(2,8,7,4)(5,11,13,16)(9,15,14,12)(17,20)(18,24)(19,26)(21,31)(22,29)(23,28)(25,30)(27,32)(33,36)(34,35)(37,45)(38,42)(39,40)(41,46)(43,48)(44,47);(1,4,6,8)(2,3,7,10)(5,12,13,15)(9,11,14,16)(17,21)(18,19)(20,30)(22,25)(23,27)(24,32)(26,28)(29,31)(33,37)(34,41)(35,43)(36,44)(38,46)(39,47)(40,45)(42,48);(1,5)(2,9)(3,11)(4,12)(6,13)(7,14)(8,15)(10,16)(17,18)(19,21)(20,24)(22,23)(25,27)(26,31)(28,29)(30,32)(33,38,39,35)(34,36,42,40)(37,46,47,43)(41,44,48,45)Polycyclic presentation:
2^9|a^2=hi,b^2=gi,c^2=i,d^2=gh,e^2=gh,f^2=i,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, 8936934);