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 | |
4483 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2)(3,8)(4,6)(5,7)(25,26,30,33)(27,34,28,31)(29,32,37,40)(35,39,36,38);(1,2,6,4)(3,8,5,7)(9,10,14,15)(11,13,17,22)(12,16,21,23)(18,20,19,24)(27,28)(31,34)(35,36)(38,39);(1,2,6,4)(3,7,5,8)(10,15)(13,22)(16,23)(20,24)(25,27,30,28)(26,31,33,34)(29,36,37,35)(32,39,40,38);(1,3,6,5)(2,7,4,8)(9,11,14,17)(10,13,15,22)(12,19,21,18)(16,24,23,20)(25,28,30,27)(26,31,33,34)(29,36,37,35)(32,38,40,39);(1,4,6,2)(3,7,5,8)(9,12)(10,16)(11,18)(13,20)(14,21)(15,23)(17,19)(22,24)(25,26)(27,31)(28,34)(29,32)(30,33)(35,38)(36,39)(37,40);(1,5,6,3)(2,7,4,8)(9,13)(10,17)(11,15)(12,20)(14,22)(16,19)(18,23)(21,24)(25,29)(26,32)(27,35)(28,36)(30,37)(31,38)(33,40)(34,39)Polycyclic presentation:
2^9|a^2=gh,b^2=hi,c^2=h,d^2=gi,e^2=g,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,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, 8202250);