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 | #38076 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4720 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2)(3,7)(4,6)(5,8)(9,10,14,17)(11,12,19,15)(13,16,21,23)(18,20,24,22)(25,26)(27,34)(28,31)(29,37)(30,33)(32,38)(35,39)(36,40);(1,2,4,6)(3,5,8,7)(9,11,14,19)(10,15,17,12)(13,18,21,24)(16,22,23,20)(27,28)(29,38)(31,34)(32,37);(1,2)(3,5)(4,6)(7,8)(13,21)(16,23)(18,24)(20,22)(25,27,30,28)(26,31,33,34)(29,35,38,36)(32,39,37,40);(1,3)(2,5)(4,8)(6,7)(9,12)(10,11)(13,20)(14,15)(16,18)(17,19)(21,22)(23,24)(25,28,30,27)(26,31,33,34)(29,36,38,35)(32,39,37,40);(9,11)(10,12)(13,18)(14,19)(15,17)(16,20)(21,24)(22,23)(25,29)(26,32)(27,35)(28,36)(30,38)(31,39)(33,37)(34,40);(9,13)(10,16)(11,18)(12,20)(14,21)(15,22)(17,23)(19,24)(25,26)(27,34)(28,31)(29,32)(30,33)(35,40)(36,39)(37,38)Polycyclic presentation:
2^9|a^2=ghi,b^2=g,c^2=h,d^2=h,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^b=egi,f^c=fgi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8109556);