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 | |
4285 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 48 points:
(1,2,6,8)(3,7,11,15)(4,9,12,14)(5,10,13,16)(17,18)(19,22)(20,30)(21,26)(23,25)(24,27)(28,31)(29,32)(34,39)(36,40)(41,47)(44,48);(1,3)(2,7)(4,5)(6,11)(8,15)(9,10)(12,13)(14,16)(17,19,23,27)(18,24,25,22)(20,28,26,32)(21,31,30,29)(33,34,38,39)(35,40,43,36)(37,41,46,47)(42,48,45,44);(2,8)(3,11)(5,13)(9,14)(18,25)(21,30)(22,24)(29,31)(33,35,38,43)(34,40,39,36)(37,45,46,42)(41,44,47,48);(1,4)(2,9)(3,5)(6,12)(7,10)(8,14)(11,13)(15,16)(17,20)(18,21)(19,28)(22,29)(23,26)(24,31)(25,30)(27,32)(33,36)(34,35)(37,44)(38,40)(39,43)(41,42)(45,47)(46,48);(1,5)(2,10)(3,12)(4,11)(6,13)(7,14)(8,16)(9,15)(17,21)(18,26)(19,29)(20,25)(22,32)(23,30)(24,28)(27,31)(33,35)(34,40)(36,39)(37,42)(38,43)(41,48)(44,47)(45,46);(1,2)(3,7)(4,9)(5,10)(6,8)(11,15)(12,14)(13,16)(17,22,23,24)(18,27,25,19)(20,29,26,31)(21,32,30,28)(33,37,38,46)(34,41,39,47)(35,42,43,45)(36,44,40,48)Polycyclic presentation:
2^9|a^2=i,b^2=g,c^2=ghi,f^2=g,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, 8850939);