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 | #57191 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4459 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)(11,12)(13,21)(14,15)(16,18)(17,22)(19,23)(20,24)(25,26,31,29)(27,32,34,30)(28,33,36,38)(35,39,40,37);(2,6)(3,7)(10,15)(12,16)(17,21)(20,23)(25,27,31,34)(26,32,29,30)(28,35,36,40)(33,39,38,37);(1,3)(2,4)(5,7)(6,8)(10,15)(11,18)(13,22)(20,23)(27,34)(28,36)(30,32)(33,38);(1,4)(2,3)(5,8)(6,7)(9,11,14,18)(10,16,15,12)(13,19,22,24)(17,23,21,20)(25,28)(26,33)(27,35)(29,38)(30,37)(31,36)(32,39)(34,40);(9,12,14,16)(10,11,15,18)(13,20,22,23)(17,19,21,24)(25,29,31,26)(27,30,34,32)(28,33,36,38)(35,39,40,37);(9,13,14,22)(10,17,15,21)(11,19,18,24)(12,20,16,23)(25,30)(26,27)(28,37)(29,34)(31,32)(33,35)(36,39)(38,40)Polycyclic presentation:
2^9|a^2=i,b^2=ghi,d^2=h,e^2=gi,f^2=h,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dgi,f^c=fgi,e^d=egi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8827937);