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 | |
4257 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 40 points:
(9,10,15,16)(11,20,21,17)(12,24,18,13)(14,19,22,23)(25,26,31,30)(27,34,33,29)(28,32,36,38)(35,40,39,37);(2,3)(6,7)(10,16)(11,21)(12,18)(19,23)(28,36)(32,38)(35,39)(37,40);(1,2,5,3)(4,7,8,6)(9,11)(10,17)(12,22)(13,23)(14,18)(15,21)(16,20)(19,24)(25,27)(26,29)(28,35)(30,34)(31,33)(32,37)(36,39)(38,40);(9,12,15,18)(10,13,16,24)(11,22,21,14)(17,23,20,19)(25,28,31,36)(26,32,30,38)(27,35,33,39)(29,37,34,40);(1,3,5,2)(4,7,8,6)(9,13)(10,18)(11,23)(12,16)(14,17)(15,24)(19,21)(20,22)(25,29,31,34)(26,33,30,27)(28,37,36,40)(32,39,38,35);(1,4,5,8)(2,6,3,7)(9,14,15,22)(10,19,16,23)(11,18,21,12)(13,17,24,20)(25,30,31,26)(27,34,33,29)(28,38,36,32)(35,40,39,37)Polycyclic presentation:
2^9|a^2=h,c^2=gi,d^2=h,e^2=hi,f^2=ghi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^b=egi,f^c=fgi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8165157);