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 |
| prime radical | #44193 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4109 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)(3,9,10,4)(5,13,15,8)(11,12,16,14)(17,18,23,20)(19,24,27,28)(21,25,31,30)(22,29,32,26)(33,34)(35,44)(36,38)(37,42)(39,43)(40,47)(41,45)(46,48);(2,7)(4,9)(8,13)(12,14)(17,19)(18,24)(20,28)(21,26)(22,25)(23,27)(29,31)(30,32)(33,35,39,47)(34,40,43,44)(36,37,45,48)(38,46,41,42);(2,7)(3,10)(8,13)(11,16)(18,20)(19,27)(25,30)(26,29)(33,36,39,45)(34,38,43,41)(35,37,47,48)(40,46,44,42);(1,3,6,10)(2,4,7,9)(5,11,15,16)(8,12,13,14)(17,20,23,18)(19,28,27,24)(21,25,31,30)(22,29,32,26)(33,36)(34,41)(35,37)(38,43)(39,45)(40,42)(44,46)(47,48);(1,4)(2,3)(5,14)(6,9)(7,10)(8,16)(11,13)(12,15)(17,21)(18,25)(19,29)(20,30)(22,28)(23,31)(24,32)(26,27)(33,37)(34,42)(35,45)(36,47)(38,40)(39,48)(41,44)(43,46);(1,5,6,15)(2,8,7,13)(3,11,10,16)(4,12,9,14)(17,22)(18,26)(19,25)(20,29)(21,24)(23,32)(27,30)(28,31)(33,38,39,41)(34,36,43,45)(35,46,47,42)(37,44,48,40)Polycyclic presentation:
2^9|a^2=h,b^2=ghi,c^2=ghi,d^2=h,f^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dgi,e^d=egi,f^e=fgi89 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 9050566);