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 | #38816 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4233 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2,5,3)(4,7,8,6)(9,10,15,17)(11,16,12,22)(13,18,19,14)(20,21,23,24)(27,34)(32,35)(36,39)(37,40);(2,3)(6,7)(9,11)(10,16)(12,15)(13,23)(14,21)(17,22)(18,24)(19,20)(25,26,30,31)(27,35,34,32)(28,29,33,38)(36,40,39,37);(2,3)(6,7)(10,17)(11,12)(13,19)(21,24)(26,31)(28,33)(32,35)(36,39);(1,2,5,3)(4,6,8,7)(9,12,15,11)(10,16,17,22)(13,23,19,20)(14,24,18,21)(25,27,30,34)(26,32,31,35)(28,36,33,39)(29,37,38,40);(1,3,5,2)(4,6,8,7)(9,13)(10,18)(11,20)(12,23)(14,17)(15,19)(16,21)(22,24)(25,28)(26,29)(27,36)(30,33)(31,38)(32,37)(34,39)(35,40);(1,4,5,8)(2,6,3,7)(9,14,15,18)(10,19,17,13)(11,21,12,24)(16,20,22,23)(25,29)(26,33)(27,37)(28,31)(30,38)(32,39)(34,40)(35,36)Polycyclic presentation:
2^9|a^2=h,b^2=ghi,d^2=gi,e^2=g,f^2=h,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,d^b=dgi,e^b=eh,d^c=dh,e^c=egi,f^e=fgi83 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 9949738);