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 | |
4389 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)(3,7)(4,8)(5,14)(6,10)(9,15)(11,13)(12,16)(17,18,23,25)(19,28,29,24)(20,30,27,22)(21,26,31,32)(33,34)(35,37)(36,44)(38,39)(40,43)(41,47)(42,48)(45,46);(1,3,6,11)(2,7,10,13)(4,5,12,15)(8,14,16,9)(17,19)(18,24)(20,21)(22,26)(23,29)(25,28)(27,31)(30,32)(34,39)(37,40)(41,44)(46,48);(3,11)(4,12)(7,13)(8,16)(18,25)(19,29)(20,27)(26,32)(33,35,38,43)(34,37,39,40)(36,42,47,45)(41,46,44,48);(1,4)(2,8)(3,5)(6,12)(7,14)(9,13)(10,16)(11,15)(17,20,23,27)(18,22,25,30)(19,21,29,31)(24,26,28,32)(33,35)(34,40)(36,45)(37,39)(38,43)(41,46)(42,47)(44,48);(1,5)(2,9)(3,12)(4,11)(6,15)(7,8)(10,14)(13,16)(17,21)(18,26)(19,27)(20,29)(22,28)(23,31)(24,30)(25,32)(33,36,38,47)(34,41,39,44)(35,42,43,45)(37,46,40,48);(1,2)(3,13)(4,8)(5,9)(6,10)(7,11)(12,16)(14,15)(17,22)(18,27)(19,26)(20,25)(21,28)(23,30)(24,31)(29,32)(33,37,38,40)(34,35,39,43)(36,46,47,48)(41,42,44,45)Polycyclic presentation:
2^9|a^2=h,b^2=hi,c^2=gh,d^2=h,e^2=gh,f^2=gh,b^a=bg,c^a=ch,d^a=dg,e^a=egi,f^a=fh,c^b=ci,e^b=eh,f^b=fgi,d^c=dghi,e^d=egi77 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 9287090);