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 | |
4557 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)(11,21)(12,17)(13,18)(14,19)(15,20)(16,22)(23,24)(25,26,31,28)(27,32,35,36)(29,39,38,33)(30,34,37,40);(1,2)(3,6)(4,7)(5,8)(9,11,15,22)(10,16,20,21)(12,23,18,14)(13,24,17,19)(25,27)(26,32)(28,36)(29,37)(30,38)(31,35)(33,34)(39,40);(1,3,5,4)(2,6,8,7)(11,22)(14,23)(16,21)(19,24)(26,28)(27,35)(30,37)(33,39);(9,12)(10,17)(11,23)(13,20)(14,22)(15,18)(16,19)(21,24)(25,28,31,26)(27,36,35,32)(29,33,38,39)(30,40,37,34);(1,4,5,3)(2,7,8,6)(9,13,15,17)(10,18,20,12)(11,24,22,19)(14,21,23,16)(25,29,31,38)(26,33,28,39)(27,37,35,30)(32,34,36,40);(9,14,15,23)(10,19,20,24)(11,12,22,18)(13,21,17,16)(25,30,31,37)(26,34,28,40)(27,38,35,29)(32,33,36,39)Polycyclic presentation:
2^9|a^2=gi,b^2=hi,c^2=ghi,d^2=h,e^2=gh,f^2=i,b^a=bg,c^a=ch,e^a=eh,c^b=ci,d^c=dh,f^c=fi,e^d=ei98 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7690426);