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 | |
4177 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 48 points:
(1,2)(3,9)(4,7)(5,13)(6,8)(10,14)(11,15)(12,16)(17,18,23,27)(19,24,28,30)(20,22,26,32)(21,25,31,29)(33,34)(35,42)(36,41)(37,38)(39,43)(40,47)(44,46)(45,48);(1,3,6,4)(2,7,8,9)(5,11,14,12)(10,15,13,16)(17,19)(18,24)(20,29)(21,22)(23,28)(25,26)(27,30)(31,32)(33,35)(34,40)(36,44)(37,45)(38,46)(39,47)(41,48)(42,43);(2,8)(7,9)(10,13)(15,16)(19,28)(21,31)(24,30)(25,29)(33,36,39,37)(34,41,43,38)(35,44,47,45)(40,48,42,46);(1,4,6,3)(2,9,8,7)(5,11,14,12)(10,15,13,16)(17,20)(18,22)(19,29)(21,24)(23,26)(25,28)(27,32)(30,31)(33,37,39,36)(34,41,43,38)(35,45,47,44)(40,48,42,46);(1,2)(3,9)(4,7)(5,10)(6,8)(11,16)(12,15)(13,14)(17,21)(18,25)(19,22)(20,30)(23,31)(24,26)(27,29)(28,32)(33,35)(34,42)(36,45)(37,44)(38,46)(39,47)(40,43)(41,48);(1,5,6,14)(2,10,8,13)(3,11,4,12)(7,15,9,16)(17,22)(18,26)(19,21)(20,27)(23,32)(24,25)(28,31)(29,30)(33,38,39,41)(34,37,43,36)(35,46,47,48)(40,45,42,44)Polycyclic presentation:
2^9|a^2=i,b^2=h,c^2=gh,d^2=g,f^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,e^c=egi,e^d=egi,f^d=fgi89 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8973226);