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 | |
4913 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)(18,23)(21,33)(24,27)(25,34)(29,43)(32,44)(36,39)(42,48);(1,3,5,7)(2,4,6,8)(9,10,13,14)(11,15,16,12)(20,31)(25,34)(28,41)(32,44)(36,39)(38,45)(42,48)(46,47);(9,11)(10,12)(13,16)(14,15)(17,18,22,23)(19,27,30,24)(20,25,31,34)(21,26,33,35)(28,39,41,36)(29,40,43,37)(32,38,44,45)(42,47,48,46);(2,6)(3,7)(10,14)(12,15)(17,19,22,30)(18,24,23,27)(20,28,31,41)(21,29,33,43)(25,36,34,39)(26,37,35,40)(32,42,44,48)(38,46,45,47);(1,2)(3,4)(5,6)(7,8)(9,12)(10,11)(13,15)(14,16)(17,20)(18,25)(19,28)(21,32)(22,31)(23,34)(24,36)(26,38)(27,39)(29,42)(30,41)(33,44)(35,45)(37,46)(40,47)(43,48);(1,4)(2,7)(3,6)(5,8)(9,11)(10,15)(12,14)(13,16)(17,21)(18,26)(19,29)(20,32)(22,33)(23,35)(24,37)(25,38)(27,40)(28,42)(30,43)(31,44)(34,45)(36,46)(39,47)(41,48)Polycyclic presentation:
2^9|b^2=gi,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, 10133261);