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 | |
4485 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)(3,8,4,6)(9,10)(11,15)(12,19)(13,14)(16,20)(17,21)(18,24)(22,23)(25,26,31,36)(27,35,30,32)(28,29,34,33)(37,38,39,40);(3,4)(6,8)(11,17)(15,21)(18,23)(22,24)(25,27,31,30)(26,32,36,35)(28,37,34,39)(29,40,33,38);(1,3,5,4)(2,6,7,8)(10,14)(15,21)(16,19)(22,24)(25,28,31,34)(26,33,36,29)(27,37,30,39)(32,38,35,40);(3,4)(6,8)(9,10)(11,15)(12,16)(13,14)(17,21)(18,22)(19,20)(23,24)(25,28)(26,33)(27,37)(29,36)(30,39)(31,34)(32,38)(35,40);(1,4,5,3)(2,6,7,8)(9,11)(10,15)(12,18)(13,17)(14,21)(16,22)(19,24)(20,23)(25,29,31,33)(26,34,36,28)(27,38,30,40)(32,37,35,39);(1,2)(3,6)(4,8)(5,7)(9,12)(10,16)(11,18)(13,20)(14,19)(15,22)(17,23)(21,24)(25,30,31,27)(26,35,36,32)(28,39,34,37)(29,38,33,40)Polycyclic presentation:
2^9|a^2=gi,b^2=g,c^2=gi,e^2=gi,f^2=g,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, 10065596);