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 |
| prime radical | #47938 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4421 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(2,5)(3,8)(9,10,13,17)(11,15,18,14)(12,20,21,16)(19,22,24,23)(29,37)(34,38)(35,36)(39,40);(1,2,6,5)(3,7,8,4)(11,18)(14,15)(19,24)(22,23)(26,31)(32,33)(34,38)(39,40);(3,8)(4,7)(9,11,13,18)(10,14,17,15)(12,19,21,24)(16,22,20,23)(25,26)(27,33)(28,32)(29,34)(30,31)(35,40)(36,39)(37,38);(1,3,6,8)(2,7,5,4)(9,11)(10,15)(12,19)(13,18)(14,17)(16,23)(20,22)(21,24)(25,27,30,28)(26,32,31,33)(29,35,37,36)(34,39,38,40);(1,4)(2,3)(5,8)(6,7)(9,10)(11,15)(12,16)(13,17)(14,18)(19,23)(20,21)(22,24)(25,28,30,27)(26,33,31,32)(29,36,37,35)(34,40,38,39);(1,5,6,2)(3,4,8,7)(9,12)(10,16)(11,19)(13,21)(14,22)(15,23)(17,20)(18,24)(25,29)(26,34)(27,35)(28,36)(30,37)(31,38)(32,39)(33,40)Polycyclic presentation:
2^9|a^2=h,b^2=g,c^2=h,d^2=ghi,e^2=hi,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,f^a=fgi,c^b=ci,d^b=dh,d^c=dgi,e^c=egi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8038382);