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 | #39854 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4392 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2,5,4)(3,7,8,6)(9,10,14,17)(11,15,18,12)(13,16,22,24)(19,23,21,20)(25,26)(27,37)(28,34)(29,40)(30,36)(31,33)(32,39)(35,38);(1,3,5,8)(2,6,4,7)(9,11)(10,15)(12,17)(13,21)(14,18)(16,20)(19,22)(23,24)(25,27)(26,32)(28,35)(29,34)(30,38)(31,39)(33,37)(36,40);(1,2,5,4)(3,6,8,7)(11,18)(12,15)(19,21)(20,23)(26,33)(28,36)(32,37)(35,40);(9,12,14,15)(10,11,17,18)(13,20,22,23)(16,19,24,21)(25,28,31,36)(26,34,33,30)(27,35,39,40)(29,37,38,32);(1,4,5,2)(3,7,8,6)(9,11)(10,15)(12,17)(13,19)(14,18)(16,23)(20,24)(21,22)(25,29,31,38)(26,35,33,40)(27,34,39,30)(28,37,36,32);(9,13)(10,16)(11,19)(12,20)(14,22)(15,23)(17,24)(18,21)(25,30,31,34)(26,36,33,28)(27,38,39,29)(32,40,37,35)Polycyclic presentation:
2^9|a^2=ghi,b^2=gh,c^2=gh,d^2=hi,e^2=g,f^2=h,b^a=bg,c^a=ch,e^a=eh,c^b=ci,f^b=fi,d^c=dhi,e^c=ei104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7643893);