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 | |
4045 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 48 points:
(1,2,6,7)(3,5,10,8)(4,12,14,9)(11,15,16,13)(17,18)(19,24)(20,22)(21,30)(23,25)(26,29)(27,32)(28,31)(33,34,38,39)(35,36,43,40)(37,41,45,46)(42,44,48,47);(2,7)(5,8)(9,12)(13,15)(17,19)(18,24)(20,29)(21,27)(22,26)(23,28)(25,31)(30,32)(34,39)(36,40)(37,45)(42,48);(2,7)(3,10)(9,12)(11,16)(18,25)(19,28)(20,27)(26,30)(33,35,38,43)(34,36,39,40)(37,42,45,48)(41,44,46,47);(1,3)(2,8)(4,11)(5,7)(6,10)(9,15)(12,13)(14,16)(17,20,23,27)(18,22,25,32)(19,29,28,21)(24,26,31,30)(33,35)(34,40)(36,39)(37,42)(38,43)(41,47)(44,46)(45,48);(1,4)(2,9)(3,11)(5,13)(6,14)(7,12)(8,15)(10,16)(17,21,23,29)(18,26,25,30)(19,27,28,20)(22,31,32,24)(33,36,38,40)(34,35,39,43)(37,44,45,47)(41,42,46,48);(1,5,6,8)(2,3,7,10)(4,13,14,15)(9,11,12,16)(17,22)(18,27)(19,30)(20,25)(21,31)(23,32)(24,29)(26,28)(33,37,38,45)(34,41,39,46)(35,42,43,48)(36,44,40,47)Polycyclic presentation:
2^9|a^2=g,c^2=ghi,d^2=i,e^2=gh,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=egi,f^a=fh,c^b=ci,e^b=eghi,f^b=fgi,d^c=dgi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8662243);