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 | #41381 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4414 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 48 points:
(1,2,6,8)(3,5,4,7)(17,18,22,26)(19,23,30,27)(20,24,31,37)(21,33,34,25)(28,35,41,39)(29,40,43,36)(32,45,44,38)(42,47,48,46);(1,3,6,4)(2,7,8,5)(9,10)(11,15)(12,14)(13,16)(20,31)(24,37)(28,41)(32,44)(35,39)(38,45)(42,48)(46,47);(9,11)(10,13)(12,16)(14,15)(17,18)(19,27)(20,24)(21,25)(22,26)(23,30)(28,39)(29,40)(31,37)(32,38)(33,34)(35,41)(36,43)(42,47)(44,45)(46,48);(2,8)(3,4)(10,14)(13,15)(17,19)(18,23)(20,28)(21,29)(22,30)(24,35)(25,36)(26,27)(31,41)(32,42)(33,40)(34,43)(37,39)(38,46)(44,48)(45,47);(1,4,6,3)(2,7,8,5)(9,10)(11,13)(12,14)(15,16)(17,20)(18,24)(19,28)(21,32)(22,31)(23,35)(25,38)(26,37)(27,39)(29,42)(30,41)(33,45)(34,44)(36,46)(40,47)(43,48);(1,5,6,7)(2,4,8,3)(9,11)(10,15)(12,16)(13,14)(17,21,22,34)(18,25,26,33)(19,29,30,43)(20,32,31,44)(23,36,27,40)(24,38,37,45)(28,42,41,48)(35,46,39,47)Polycyclic presentation:
2^9|a^2=gh,b^2=g,e^2=g,f^2=gh,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, 10127028);