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 | |
4137 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 48 points:
(1,2,6,10)(3,7,11,8)(4,5,9,15)(12,13,14,16)(19,26)(21,29)(23,24)(25,28)(33,34,39,44)(35,42,37,40)(36,46,47,41)(38,43,45,48);(1,3)(2,7)(4,12)(5,13)(6,11)(8,10)(9,14)(15,16)(17,18)(19,24)(20,22)(21,25)(23,26)(27,31)(28,29)(30,32)(33,35,39,37)(34,40,44,42)(36,38,47,45)(41,43,46,48);(3,11)(7,8)(12,14)(13,16)(17,19,22,26)(18,23,20,24)(21,27,29,32)(25,30,28,31)(33,36,39,47)(34,41,44,46)(35,45,37,38)(40,48,42,43);(1,3)(2,8)(4,14)(5,13)(6,11)(7,10)(9,12)(15,16)(17,19)(18,24)(20,23)(21,27)(22,26)(25,31)(28,30)(29,32)(33,37,39,35)(34,42,44,40)(36,38,47,45)(41,43,46,48);(1,4)(2,5)(3,12)(6,9)(7,13)(8,16)(10,15)(11,14)(17,20,22,18)(19,24,26,23)(21,25,29,28)(27,30,32,31)(33,38)(34,43)(35,36)(37,47)(39,45)(40,41)(42,46)(44,48);(1,5)(2,9)(3,13)(4,10)(6,15)(7,14)(8,12)(11,16)(17,21)(18,25)(19,27)(20,28)(22,29)(23,30)(24,31)(26,32)(33,34)(35,40)(36,41)(37,42)(38,43)(39,44)(45,48)(46,47)Polycyclic presentation:
2^9|a^2=ghi,b^2=g,c^2=h,d^2=g,e^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, 9078853);