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 | |
4340 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(9,10,14,16)(11,17,18,12)(13,15,21,22)(19,23,24,20)(25,26)(27,33)(28,32)(29,30)(31,34)(35,40)(36,39)(37,38);(1,2,5,4)(3,7,8,6)(9,10)(11,12)(13,15)(14,16)(17,18)(19,20)(21,22)(23,24)(27,34)(31,33)(35,36)(39,40);(1,3,5,8)(2,6,4,7)(11,18)(12,17)(13,21)(15,22)(25,27,30,34)(26,31,29,33)(28,36,38,35)(32,40,37,39);(3,8)(6,7)(9,11,14,18)(10,12,16,17)(13,19,21,24)(15,20,22,23)(25,28,30,38)(26,32,29,37)(27,35,34,36)(31,39,33,40);(1,3)(2,7)(4,6)(5,8)(9,12)(10,11)(13,20)(14,17)(15,19)(16,18)(21,23)(22,24)(25,27)(26,33)(28,36)(29,31)(30,34)(32,39)(35,38)(37,40);(1,4,5,2)(3,6,8,7)(9,13,14,21)(10,15,16,22)(11,19,18,24)(12,20,17,23)(25,29,30,26)(27,31,34,33)(28,37,38,32)(35,39,36,40)Polycyclic presentation:
2^9|a^2=g,b^2=hi,c^2=i,d^2=gh,f^2=gi,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, 8935301);