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 | |
4212 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 48 points:
(1,2,7,4)(3,8,10,6)(5,9,13,12)(11,15,16,14)(17,18)(19,21)(20,29)(22,30)(23,25)(24,28)(26,27)(31,32)(33,34,39,43)(35,42,47,40)(36,41,37,38)(44,46,45,48);(1,3,7,10)(2,8,4,6)(5,11,13,16)(9,15,12,14)(17,19,23,28)(18,24,25,21)(20,26,30,32)(22,27,29,31)(33,35)(34,40)(36,44)(37,45)(38,46)(39,47)(41,48)(42,43);(3,10)(5,13)(6,8)(9,12)(18,25)(21,24)(22,29)(27,31)(33,36,39,37)(34,41,43,38)(35,44,47,45)(40,48,42,46);(1,4,7,2)(3,6,10,8)(5,9,13,12)(11,15,16,14)(17,20)(18,22)(19,26)(21,31)(23,30)(24,27)(25,29)(28,32)(33,37,39,36)(34,41,43,38)(35,45,47,44)(40,48,42,46);(1,5)(2,9)(3,11)(4,12)(6,14)(7,13)(8,15)(10,16)(17,21,23,24)(18,19,25,28)(20,27,30,31)(22,32,29,26)(33,35)(34,42)(36,45)(37,44)(38,46)(39,47)(40,43)(41,48);(1,6)(2,3)(4,10)(5,14)(7,8)(9,11)(12,16)(13,15)(17,22)(18,20)(19,27)(21,32)(23,29)(24,26)(25,30)(28,31)(33,38,39,41)(34,36,43,37)(35,46,47,48)(40,44,42,45)Polycyclic presentation:
2^9|a^2=ghi,b^2=hi,c^2=gh,d^2=ghi,e^2=h,f^2=gh,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,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, 8864800);