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 | |
4530 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)(3,5)(4,6)(7,8)(10,15)(11,18)(17,24)(19,20)(25,26)(27,35)(28,30)(29,32)(31,34)(33,40)(36,37)(38,39);(1,3,7,6)(2,4,8,5)(9,10,14,15)(11,16,18,13)(12,17,22,24)(19,21,20,23)(27,29)(32,35)(36,38)(37,39);(1,4,7,5)(2,3,8,6)(25,27,31,29)(26,32,34,35)(28,38,33,36)(30,37,40,39);(2,8)(4,5)(9,11)(10,16)(12,20)(13,15)(14,18)(17,23)(19,22)(21,24)(25,28,31,33)(26,30,34,40)(27,36,29,38)(32,39,35,37);(1,5,7,4)(2,3,8,6)(9,12,14,22)(10,17,15,24)(11,19,18,20)(13,23,16,21)(25,29,31,27)(26,32,34,35)(28,36,33,38)(30,37,40,39);(1,6,7,3)(2,5,8,4)(9,13,14,16)(10,11,15,18)(12,21,22,23)(17,20,24,19)(25,30)(26,33)(27,37)(28,34)(29,39)(31,40)(32,36)(35,38)Polycyclic presentation:
2^9|b^2=g,c^2=h,d^2=i,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, 10073727);