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 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4688 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(9,10,14,19)(11,15,20,16)(12,17,22,23)(13,21,24,18)(25,26,31,30)(27,34,29,32)(28,36,38,33)(35,39,37,40);(1,2,5,7)(3,4,8,6)(9,11)(10,15)(12,21)(13,23)(14,20)(16,19)(17,24)(18,22)(25,27,31,29)(26,32,30,34)(28,37,38,35)(33,40,36,39);(1,3,5,8)(2,6,7,4)(9,11,14,20)(10,16,19,15)(12,18,22,21)(13,17,24,23)(25,26,31,30)(27,32,29,34)(28,33,38,36)(35,39,37,40);(1,3,5,8)(2,6,7,4)(9,12)(10,17)(11,18)(13,15)(14,22)(16,24)(19,23)(20,21)(25,28,31,38)(26,33,30,36)(27,35,29,37)(32,39,34,40);(1,3)(2,4)(5,8)(6,7)(9,11)(10,15)(12,18)(13,17)(14,20)(16,19)(21,22)(23,24)(25,29,31,27)(26,32,30,34)(28,35,38,37)(33,40,36,39);(1,4)(2,3)(5,6)(7,8)(9,13)(10,18)(11,17)(12,15)(14,24)(16,22)(19,21)(20,23)(25,30,31,26)(27,32,29,34)(28,36,38,33)(35,39,37,40)Polycyclic presentation:
2^9|a^2=gh,b^2=ghi,c^2=gi,d^2=ghi,e^2=g,f^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,d^b=dgi,f^b=fgi,e^c=egi,e^d=eghi83 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 10273459);