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 | |
4531 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 40 points:
(9,10,14,16)(11,19,20,15)(12,21,22,17)(13,18,23,24)(25,26,30,32)(27,33,34,29)(28,31,36,39)(35,40,37,38);(1,2,5,4)(3,7,8,6)(9,11,14,20)(10,15,16,19)(12,13,22,23)(17,18,21,24)(25,26)(27,29)(28,31)(30,32)(33,34)(35,38)(36,39)(37,40);(1,3,5,8)(2,6,4,7)(10,16)(12,22)(13,23)(15,19)(27,34)(28,36)(29,33)(31,39);(3,8)(6,7)(9,12,14,22)(10,17,16,21)(11,13,20,23)(15,18,19,24)(25,27)(26,29)(28,37)(30,34)(31,40)(32,33)(35,36)(38,39);(1,3)(2,7)(4,6)(5,8)(9,13)(10,18)(11,12)(14,23)(15,17)(16,24)(19,21)(20,22)(25,28)(26,31)(27,35)(29,38)(30,36)(32,39)(33,40)(34,37);(1,4,5,2)(3,6,8,7)(9,10)(11,15)(12,17)(13,18)(14,16)(19,20)(21,22)(23,24)(25,29)(26,27)(28,38)(30,33)(31,35)(32,34)(36,40)(37,39)Polycyclic presentation:
2^9|a^2=g,b^2=hi,c^2=i,d^2=h,f^2=i,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dgi,e^c=egi,f^c=fgi,e^d=egi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8935261);