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 | |
4461 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)(11,17)(15,18)(19,23)(22,24)(25,26)(27,36)(28,33)(29,34)(30,32)(31,35)(37,40)(38,39);(2,3)(6,7)(9,10)(11,18)(12,16)(13,14)(15,17)(19,24)(20,21)(22,23)(25,27,31,30)(26,32,35,36)(28,37,29,38)(33,39,34,40);(2,3)(6,7)(10,14)(15,18)(16,20)(22,24)(25,28,31,29)(26,33,35,34)(27,37,30,38)(32,39,36,40);(1,2,5,3)(4,6,8,7)(9,11)(10,15)(12,19)(13,17)(14,18)(16,22)(20,24)(21,23)(25,29,31,28)(26,34,35,33)(27,37,30,38)(32,39,36,40);(1,3,5,2)(4,6,8,7)(9,10)(11,15)(12,20)(13,14)(16,21)(17,18)(19,24)(22,23)(25,30,31,27)(26,32,35,36)(28,37,29,38)(33,40,34,39);(1,4)(2,6)(3,7)(5,8)(9,12)(10,16)(11,19)(13,21)(14,20)(15,22)(17,23)(18,24)(25,26)(27,32)(28,33)(29,34)(30,36)(31,35)(37,39)(38,40)Polycyclic presentation:
2^9|a^2=h,b^2=gh,c^2=gh,d^2=g,e^2=g,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,d^b=dgi,e^b=eh,d^c=dh,e^c=egi,f^e=fgi83 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 9928004);