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 | |
4489 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2,5,8)(3,6,4,7)(9,10)(11,12)(13,21)(14,16)(15,19)(17,22)(18,23)(20,24)(25,26)(27,33)(28,32)(29,31)(30,34)(35,40)(36,39)(37,38);(1,2)(3,6)(4,7)(5,8)(9,11,14,19)(10,15,16,12)(13,18,22,24)(17,23,21,20)(25,27)(26,30)(28,36)(29,34)(31,33)(32,40)(35,38)(37,39);(1,3,5,4)(2,6,8,7)(10,16)(12,15)(17,21)(20,23)(26,31)(27,34)(32,37)(35,36);(1,3,5,4)(2,6,8,7)(9,12)(10,11)(13,20)(14,15)(16,19)(17,18)(21,24)(22,23)(25,26)(27,30)(28,37)(29,31)(32,38)(33,34)(35,40)(36,39);(1,3,5,4)(2,6,8,7)(9,11)(10,12)(13,18)(14,19)(15,16)(17,20)(21,23)(22,24)(25,28,29,38)(26,32,31,37)(27,35,34,36)(30,39,33,40);(1,4,5,3)(2,7,8,6)(9,13)(10,17)(11,18)(12,20)(14,22)(15,23)(16,21)(19,24)(25,27)(26,30)(28,35)(29,34)(31,33)(32,39)(36,38)(37,40)Polycyclic presentation:
2^9|a^2=gh,b^2=hi,c^2=gh,d^2=gh,e^2=ghi,f^2=gh,b^a=bg,c^a=ch,f^a=fh,c^b=ci,e^b=eh,d^c=dh,f^c=fi,e^d=ei92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7919273);