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 | #38034 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4391 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)(9,10,14,15)(11,13,18,16)(12,20,21,17)(19,23,24,22);(1,2,4,6)(3,5,8,7)(10,15)(11,18)(12,21)(22,23)(25,26,31,30)(27,34,28,32)(29,37,38,33)(35,39,36,40);(1,2)(3,5)(4,6)(7,8)(10,15)(13,16)(17,20)(22,23)(25,27,31,28)(26,32,30,34)(29,35,38,36)(33,39,37,40);(1,3,4,8)(2,5,6,7)(9,11,14,18)(10,16,15,13)(12,19,21,24)(17,23,20,22)(25,28,31,27)(26,32,30,34)(29,36,38,35)(33,39,37,40);(9,12)(10,17)(11,19)(13,22)(14,21)(15,20)(16,23)(18,24)(25,29,31,38)(26,33,30,37)(27,35,28,36)(32,39,34,40);(9,13,14,16)(10,18,15,11)(12,22,21,23)(17,24,20,19)(25,30,31,26)(27,32,28,34)(29,37,38,33)(35,39,36,40)Polycyclic presentation:
2^9|a^2=h,b^2=i,c^2=ghi,d^2=hi,e^2=ghi,f^2=gi,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, 8178373);