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 | #59947 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4524 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2,5,6)(3,4,7,8)(9,10)(11,15)(12,21)(13,18)(14,19)(16,20)(17,23)(22,24)(25,26,31,35)(27,32,37,40)(28,30,39,33)(29,38,36,34);(2,6)(3,7)(9,11,14,20)(10,15,19,16)(12,22,23,18)(13,21,24,17)(25,27)(26,32)(28,38)(29,33)(30,36)(31,37)(34,39)(35,40);(1,3)(2,4)(5,7)(6,8)(11,20)(13,24)(15,16)(18,22)(25,28,31,39)(26,33,35,30)(27,34,37,38)(29,32,36,40);(1,4)(2,3)(5,8)(6,7)(9,11)(10,16)(12,22)(13,21)(14,20)(15,19)(17,24)(18,23)(25,28)(26,33)(27,34)(29,40)(30,35)(31,39)(32,36)(37,38);(9,12,14,23)(10,17,19,21)(11,18,20,22)(13,15,24,16)(25,29,31,36)(26,34,35,38)(27,33,37,30)(28,32,39,40);(9,13)(10,18)(11,21)(12,15)(14,24)(16,23)(17,20)(19,22)(25,30,31,33)(26,28,35,39)(27,36,37,29)(32,38,40,34)Polycyclic presentation:
2^9|a^2=gh,b^2=ghi,c^2=h,e^2=gi,f^2=h,b^a=bg,c^a=ch,d^a=di,e^a=egi,f^a=fh,c^b=ci,d^b=dgi,e^b=eghi,d^c=dgi,f^c=fgi,e^d=egi92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8842075);