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 | |
4261 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2,5,7)(3,4,8,6)(10,15)(11,18)(13,21)(20,23)(25,26,31,27)(28,35,38,32)(29,33,30,34)(36,40,37,39);(1,2)(3,6)(4,8)(5,7)(9,10,14,15)(11,12,18,16)(13,17,21,22)(19,20,24,23)(26,27)(32,35)(33,34)(39,40);(1,3,5,8)(2,6,7,4)(9,11,14,18)(10,16,15,12)(13,19,21,24)(17,23,22,20)(25,27,31,26)(28,35,38,32)(29,34,30,33)(36,40,37,39);(1,2)(3,4)(5,7)(6,8)(9,11)(10,12)(13,19)(14,18)(15,16)(17,20)(21,24)(22,23)(25,28,31,38)(26,32,27,35)(29,36,30,37)(33,39,34,40);(1,3)(2,4)(5,8)(6,7)(9,12)(10,11)(13,20)(14,16)(15,18)(17,19)(21,23)(22,24)(25,29,31,30)(26,33,27,34)(28,36,38,37)(32,39,35,40);(1,4)(2,3)(5,6)(7,8)(9,13)(10,17)(11,19)(12,20)(14,21)(15,22)(16,23)(18,24)(25,30,31,29)(26,34,27,33)(28,37,38,36)(32,40,35,39)Polycyclic presentation:
2^9|a^2=hi,b^2=ghi,c^2=g,d^2=gh,e^2=gh,f^2=gh,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,f^b=fgi,d^c=dh,e^c=egi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 9152344);