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 | |
3997 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)(3,4)(5,13)(6,8)(7,15)(9,11)(10,14)(12,16)(17,18)(19,25)(20,29)(21,23)(22,26)(24,27)(28,30)(31,32)(33,34,39,38)(35,43,45,40)(36,41,44,46)(37,42,47,48);(1,3)(2,4)(5,10)(6,11)(7,12)(8,9)(13,14)(15,16)(17,19,24,21)(18,25,27,23)(20,28,22,31)(26,32,29,30)(33,35,39,45)(34,40,38,43)(36,37,44,47)(41,48,46,42);(1,2,6,8)(3,9,11,4)(5,7,15,13)(10,14,16,12)(17,20,24,22)(18,26,27,29)(19,28,21,31)(23,30,25,32)(33,36,39,44)(34,41,38,46)(35,37,45,47)(40,48,43,42);(1,2)(3,9)(4,11)(5,7)(6,8)(10,14)(12,16)(13,15)(17,21,24,19)(18,23,27,25)(20,28,22,31)(26,32,29,30)(33,35)(34,40)(36,37)(38,43)(39,45)(41,48)(42,46)(44,47);(1,4,6,9)(2,3,8,11)(5,14,15,12)(7,16,13,10)(17,22,24,20)(18,26,27,29)(19,28,21,31)(23,32,25,30)(33,37)(34,42)(35,44)(36,45)(38,48)(39,47)(40,41)(43,46);(1,5)(2,7)(3,10)(4,12)(6,15)(8,13)(9,14)(11,16)(17,23)(18,19)(20,30)(21,27)(22,32)(24,25)(26,28)(29,31)(33,38,39,34)(35,43,45,40)(36,46,44,41)(37,42,47,48)Polycyclic presentation:
2^9|a^2=g,b^2=ghi,c^2=gh,d^2=hi,e^2=h,f^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,d^b=dgi,e^b=eh,d^c=dh,e^d=egi,f^e=fgi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 10154499);