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 | #38628 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4307 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2,5,3)(4,7,8,6)(9,10)(11,16)(12,15)(13,17)(14,20)(18,23)(19,21)(22,24)(25,26)(27,28)(29,37)(30,31)(32,35)(33,38)(34,39)(36,40);(2,3)(6,7)(9,11,15,17)(10,16,12,13)(14,19,23,24)(18,22,20,21)(26,31)(28,32)(33,37)(36,39);(1,3,5,2)(4,7,8,6)(10,12)(11,17)(18,20)(19,24)(25,26,30,31)(27,32,35,28)(29,33,38,37)(34,39,40,36);(1,4,5,8)(2,6,3,7)(9,12,15,10)(11,13,17,16)(14,20,23,18)(19,21,24,22)(25,27)(26,32)(28,31)(29,34)(30,35)(33,39)(36,37)(38,40);(9,13)(10,17)(11,12)(14,22)(15,16)(18,19)(20,24)(21,23)(25,28,30,32)(26,27,31,35)(29,36,38,39)(33,34,37,40);(9,14,15,23)(10,18,12,20)(11,19,17,24)(13,21,16,22)(25,29,30,38)(26,33,31,37)(27,34,35,40)(28,36,32,39)Polycyclic presentation:
2^9|a^2=hi,b^2=gi,c^2=g,d^2=gh,e^2=ghi,f^2=h,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eh,d^c=dgi,e^d=egi,f^e=fgi83 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 9113758);