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 | |
4662 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)(3,7)(4,5)(6,8)(9,10,14,12)(11,15,18,16)(13,17,21,20)(19,22,24,23)(27,34)(29,37)(31,32)(33,36);(2,5)(6,7)(9,11)(10,15)(12,16)(13,19)(14,18)(17,22)(20,23)(21,24)(25,26)(27,32)(28,30)(29,33)(31,34)(35,39)(36,37)(38,40);(1,2)(3,6)(4,5)(7,8)(11,18)(15,16)(19,24)(22,23)(25,27,30,34)(26,31,28,32)(29,35,37,40)(33,38,36,39);(1,3,4,8)(2,6,5,7)(9,11)(10,16)(12,15)(13,19)(14,18)(17,23)(20,22)(21,24)(25,27)(26,32)(28,31)(29,35)(30,34)(33,39)(36,38)(37,40);(9,12,14,10)(11,16,18,15)(13,17,21,20)(19,22,24,23)(25,28,30,26)(27,32,34,31)(29,33,37,36)(35,38,40,39);(9,13)(10,17)(11,19)(12,20)(14,21)(15,22)(16,23)(18,24)(25,29)(26,33)(27,35)(28,36)(30,37)(31,38)(32,39)(34,40)Polycyclic presentation:
2^9|a^2=ghi,c^2=h,d^2=g,e^2=gi,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, 9110983);