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 | |
4491 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 48 points:
(9,10)(11,15)(12,13)(14,16)(17,18,24,21)(19,25,20,28)(22,26,36,33)(23,34,37,27)(29,38,31,40)(30,42,32,39)(35,45,46,41)(43,48,44,47);(1,2)(3,7)(4,6)(5,8)(10,13)(14,15)(17,19,24,20)(18,25,21,28)(22,31,36,29)(23,30,37,32)(26,40,33,38)(27,39,34,42)(35,44,46,43)(41,48,45,47);(1,3)(2,5)(4,8)(6,7)(9,10)(11,14)(12,13)(15,16)(17,20,24,19)(18,25,21,28)(22,31,36,29)(23,32,37,30)(26,38,33,40)(27,39,34,42)(35,44,46,43)(41,47,45,48);(2,6)(5,7)(9,11)(10,14)(12,16)(13,15)(17,21,24,18)(19,28,20,25)(22,33,36,26)(23,34,37,27)(29,40,31,38)(30,42,32,39)(35,45,46,41)(43,48,44,47);(1,2)(3,5)(4,6)(7,8)(17,22)(18,26)(19,29)(20,31)(21,33)(23,35)(24,36)(25,38)(27,41)(28,40)(30,43)(32,44)(34,45)(37,46)(39,47)(42,48);(1,3)(2,7)(4,8)(5,6)(17,23)(18,27)(19,30)(20,32)(21,34)(22,35)(24,37)(25,39)(26,41)(28,42)(29,43)(31,44)(33,45)(36,46)(38,47)(40,48)Polycyclic presentation:
2^9|a^2=h,b^2=h,c^2=h,d^2=h,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, 10125852);