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 | |
4728 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)(11,13)(12,16)(14,15)(17,21)(18,20)(19,22)(23,24)(25,26)(27,31)(28,33)(29,34)(30,32)(35,38)(36,40)(37,39);(2,3)(6,7)(12,19)(16,22)(18,23)(20,24)(25,27,30,34)(26,31,32,29)(28,36,39,35)(33,40,37,38);(1,3,5,2)(4,7,8,6)(10,15)(11,17)(16,22)(18,23)(26,32)(27,34)(33,37)(35,36);(1,4)(2,6)(3,7)(5,8)(9,11,14,17)(10,13,15,21)(12,18,19,23)(16,20,22,24)(25,26)(27,29)(28,37)(30,32)(31,34)(33,39)(35,38)(36,40);(9,12,14,19)(10,16,15,22)(11,18,17,23)(13,20,21,24)(25,28,30,39)(26,33,32,37)(27,35,34,36)(29,40,31,38);(9,13)(10,17)(11,15)(12,20)(14,21)(16,23)(18,22)(19,24)(25,29)(26,27)(28,38)(30,31)(32,34)(33,36)(35,37)(39,40)Polycyclic presentation:
2^9|a^2=g,b^2=gi,c^2=g,d^2=ghi,e^2=h,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,d^b=dgi,e^b=eh,d^c=dh,e^d=egi,f^e=fgi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 10154270);