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 | |
4179 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)(3,8)(4,6)(5,12)(7,11)(9,14)(10,15)(13,16)(17,18,23,27)(19,24,28,32)(20,25,29,30)(21,22,26,31)(33,34,39,44)(35,45,48,40)(36,41,47,42)(37,43,46,38);(1,3,6,11)(2,7,4,8)(5,10,14,13)(9,15,12,16)(17,19,23,28)(18,24,27,32)(20,21,29,26)(22,30,31,25)(33,35)(34,40)(36,46)(37,47)(38,42)(39,48)(41,43)(44,45);(2,4)(7,8)(9,12)(15,16)(19,28)(20,29)(24,32)(25,30)(33,36,39,47)(34,41,44,42)(35,46,48,37)(38,40,43,45);(1,4,6,2)(3,8,11,7)(5,12,14,9)(10,16,13,15)(17,20)(18,25)(19,21)(22,24)(23,29)(26,28)(27,30)(31,32)(33,36)(34,42)(35,46)(37,48)(38,40)(39,47)(41,44)(43,45);(1,3)(2,8)(4,7)(5,13)(6,11)(9,15)(10,14)(12,16)(17,21)(18,22)(19,29)(20,28)(23,26)(24,30)(25,32)(27,31)(33,37,39,46)(34,43,44,38)(35,47,48,36)(40,41,45,42);(1,5)(2,9)(3,10)(4,12)(6,14)(7,15)(8,16)(11,13)(17,22)(18,26)(19,30)(20,24)(21,27)(23,31)(25,28)(29,32)(33,38,39,43)(34,37,44,46)(35,42,48,41)(36,40,47,45)Polycyclic presentation:
2^9|a^2=ghi,b^2=hi,c^2=gh,d^2=h,e^2=gh,f^2=gh,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dgi,e^d=egi,f^e=fgi89 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 9044106);