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 | |
4676 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)(3,4)(5,7)(6,8)(9,10)(11,18)(12,16)(13,21)(14,15)(17,23)(19,20)(22,24)(25,26,31,33)(27,36,37,32)(28,38,35,30)(29,34,39,40);(1,3,6,5)(2,7,8,4)(10,15)(12,18)(17,21)(20,22)(25,27)(26,32)(28,29)(30,34)(31,37)(33,36)(35,39)(38,40);(10,15)(11,16)(17,21)(19,24)(26,33)(27,37)(28,35)(34,40);(1,4,6,7)(2,5,8,3)(9,11)(10,12)(13,19)(14,16)(15,18)(17,20)(21,22)(23,24)(25,28,31,35)(26,30,33,38)(27,29,37,39)(32,34,36,40);(1,5,6,3)(2,7,8,4)(9,12)(10,16)(11,15)(13,22)(14,18)(17,19)(20,23)(21,24)(25,29)(26,34)(27,35)(28,37)(30,36)(31,39)(32,38)(33,40);(1,2)(3,7)(4,5)(6,8)(9,13,14,23)(10,17,15,21)(11,19,16,24)(12,20,18,22)(25,30)(26,35)(27,34)(28,33)(29,36)(31,38)(32,39)(37,40)Polycyclic presentation:
2^9|a^2=i,b^2=gh,d^2=ghi,e^2=gh,f^2=hi,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dh,e^d=egi,f^e=fgi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 10109198);