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 | #2673 | C22 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4211 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)(3,7)(4,5)(6,8)(10,11)(15,17)(16,18)(23,24)(25,26)(27,35)(28,33)(29,34)(30,31)(32,36)(37,40)(38,39);(2,5)(6,7)(9,10,14,11)(12,15,20,17)(13,16,22,18)(19,23,21,24)(27,36)(29,37)(32,35)(34,40);(1,2)(3,6)(4,5)(7,8)(25,27,31,36)(26,32,30,35)(28,37,39,29)(33,34,38,40);(1,3)(2,6)(4,8)(5,7)(9,11,14,10)(12,15,20,17)(13,18,22,16)(19,23,21,24)(25,28,31,39)(26,33,30,38)(27,29,36,37)(32,40,35,34);(9,12,14,20)(10,15,11,17)(13,21,22,19)(16,24,18,23)(25,29)(26,34)(27,28)(30,40)(31,37)(32,38)(33,35)(36,39);(9,13,14,22)(10,16,11,18)(12,19,20,21)(15,23,17,24)(25,30,31,26)(27,35,36,32)(28,38,39,33)(29,34,37,40)Polycyclic presentation:
2^9|b^2=ghi,c^2=h,d^2=gi,e^2=ghi,f^2=gi,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dh,e^d=egi,f^e=fgi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 10123857);