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 | |
4905 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 48 points:
(1,2)(3,8)(4,7)(5,6)(17,18,23,24)(19,20,28,25)(21,26,33,37)(22,35,36,27)(29,31,41,38)(30,39,42,32)(34,46,45,40)(43,47,48,44);(1,3,5,7)(2,4,6,8)(9,10)(11,15)(12,14)(13,16)(18,24)(19,28)(21,33)(27,35)(30,42)(31,38)(34,45)(44,47);(9,11)(10,13)(12,16)(14,15)(17,19,23,28)(18,25,24,20)(21,29,33,41)(22,30,36,42)(26,38,37,31)(27,39,35,32)(34,43,45,48)(40,47,46,44);(2,6)(3,7)(10,14)(13,15)(17,20,23,25)(18,19,24,28)(21,31,33,38)(22,32,36,39)(26,29,37,41)(27,30,35,42)(34,44,45,47)(40,43,46,48);(1,2)(3,4)(5,6)(7,8)(9,10)(11,13)(12,14)(15,16)(17,21)(18,26)(19,29)(20,31)(22,34)(23,33)(24,37)(25,38)(27,40)(28,41)(30,43)(32,44)(35,46)(36,45)(39,47)(42,48);(1,4)(2,7)(3,6)(5,8)(9,11)(10,15)(12,16)(13,14)(17,22)(18,27)(19,30)(20,32)(21,34)(23,36)(24,35)(25,39)(26,40)(28,42)(29,43)(31,44)(33,45)(37,46)(38,47)(41,48)Polycyclic presentation:
2^9|a^2=h,b^2=gh,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, 10125854);