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 | |
3943 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)(3,6)(4,8)(5,9)(7,11)(10,14)(12,13)(15,16)(17,18)(19,26)(20,23)(21,29)(22,25)(24,30)(27,28)(31,32)(33,34,39,36)(35,42,38,40)(37,41,46,44)(43,48,45,47);(1,3)(2,6)(4,10)(5,11)(7,9)(8,14)(12,15)(13,16)(17,18,22,25)(19,20,23,26)(21,24,30,29)(27,28,31,32)(33,34,39,36)(35,40,38,42)(37,44,46,41)(43,48,45,47);(1,3,5,11)(2,6,9,7)(4,10,13,16)(8,14,12,15)(17,19,22,23)(18,20,25,26)(21,27,30,31)(24,28,29,32)(33,35,39,38)(34,40,36,42)(37,43,46,45)(41,47,44,48);(1,3)(2,7)(4,10)(5,11)(6,9)(8,15)(12,14)(13,16)(17,20,22,26)(18,23,25,19)(21,28,30,32)(24,31,29,27)(33,36,39,34)(35,42,38,40)(37,44,46,41)(43,48,45,47);(1,2)(3,6)(4,12)(5,9)(7,11)(8,13)(10,15)(14,16)(17,18)(19,20)(21,29)(22,25)(23,26)(24,30)(27,32)(28,31)(33,37,39,46)(34,41,36,44)(35,43,38,45)(40,47,42,48);(1,4)(2,8)(3,10)(5,13)(6,14)(7,15)(9,12)(11,16)(17,21)(18,24)(19,27)(20,28)(22,30)(23,31)(25,29)(26,32)(33,38,39,35)(34,42,36,40)(37,45,46,43)(41,48,44,47)Polycyclic presentation:
2^9|a^2=gh,b^2=h,c^2=hi,d^2=h,e^2=gh,f^2=gh,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eh,d^c=dgi,e^d=egi,f^e=fgi83 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 9091104);