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 | |
4116 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 48 points:
(17,18)(19,21)(20,25)(22,29)(23,24)(26,31)(27,30)(28,32)(33,34)(35,44)(36,39)(37,47)(38,42)(40,43)(41,46)(45,48);(1,2)(3,7)(4,8)(5,6)(9,15)(10,13)(11,16)(12,14)(17,19)(18,21)(20,27)(22,28)(23,26)(24,31)(25,30)(29,32)(33,35)(34,41)(36,45)(37,42)(38,39)(40,46)(43,44)(47,48);(2,6)(7,13)(8,14)(15,16)(18,24)(19,26)(22,25)(27,32)(33,36,40,37)(34,39,43,47)(35,45,46,42)(38,44,48,41);(1,3,5,10)(2,7,6,13)(4,11,12,9)(8,16,14,15)(17,20)(18,25)(19,27)(21,30)(22,24)(23,29)(26,32)(28,31)(33,37,40,36)(34,39,43,47)(35,42,46,45)(38,44,48,41);(1,4,5,12)(2,8,6,14)(3,9,10,11)(7,15,13,16)(17,21,23,31)(18,19,24,26)(20,28,29,30)(22,27,25,32)(33,38)(34,42)(35,39)(36,44)(37,41)(40,48)(43,45)(46,47);(1,2)(3,7)(4,8)(5,6)(9,15)(10,13)(11,16)(12,14)(17,22,23,25)(18,20,24,29)(19,28,26,30)(21,27,31,32)(33,39)(34,36)(35,38)(37,43)(40,47)(41,45)(42,44)(46,48)Polycyclic presentation:
2^9|c^2=g,d^2=ghi,e^2=i,f^2=h,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,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, 8808770);