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 | |
4164 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)(4,9)(5,8)(7,12)(10,14)(11,16)(13,15)(17,18)(19,26)(20,28)(21,23)(22,25)(24,29)(27,31)(30,32)(33,34,39,42)(35,38,40,36)(37,41,47,48)(43,46,45,44);(2,3)(7,9)(8,10)(15,16)(17,19,22,21)(18,23,25,26)(20,27,29,30)(24,31,28,32)(33,35)(34,38)(36,42)(37,45)(39,40)(41,44)(43,47)(46,48);(2,3)(7,9)(8,10)(15,16)(20,29)(24,28)(27,30)(31,32)(33,36,39,38)(34,40,42,35)(37,46,47,44)(41,43,48,45);(1,3,6,2)(4,9,12,7)(5,8,14,10)(11,15,13,16)(17,20,22,29)(18,24,25,28)(19,27,21,30)(23,31,26,32)(33,37,39,47)(34,41,42,48)(35,43,40,45)(36,44,38,46);(1,4,6,12)(2,7,3,9)(5,13,14,11)(8,16,10,15)(17,21,22,19)(18,23,25,26)(20,30,29,27)(24,31,28,32)(33,35)(34,36)(37,43)(38,42)(39,40)(41,44)(45,47)(46,48);(1,5,6,14)(2,8,3,10)(4,11,12,13)(7,15,9,16)(17,18)(19,23)(20,28)(21,26)(22,25)(24,29)(27,32)(30,31)(33,38,39,36)(34,40,42,35)(37,44,47,46)(41,43,48,45)Polycyclic presentation:
2^9|a^2=i,b^2=ghi,c^2=i,d^2=gi,e^2=g,f^2=h,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,d^c=dgi,f^d=fgi,f^e=fgi89 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8627881);