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 | |
4573 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)(3,5)(4,7)(6,8)(10,15)(11,17)(16,22)(18,23)(25,26,31,34)(27,32,35,29)(28,30,33,39)(36,37,40,38);(2,3)(6,7)(9,10,14,15)(11,13,17,21)(12,16,19,22)(18,20,23,24)(25,27,31,35)(26,32,34,29)(28,36,33,40)(30,37,39,38);(1,2,5,3)(4,6,8,7)(12,19)(16,22)(18,23)(20,24)(27,35)(29,32)(36,40)(37,38);(1,3,5,2)(4,7,8,6)(9,11)(10,13)(12,18)(14,17)(15,21)(16,20)(19,23)(22,24)(25,28)(26,30)(27,36)(29,38)(31,33)(32,37)(34,39)(35,40);(1,4,5,8)(2,6,3,7)(9,12,14,19)(10,16,15,22)(11,18,17,23)(13,20,21,24)(25,29)(26,27)(28,38)(30,36)(31,32)(33,37)(34,35)(39,40);(9,13)(10,17)(11,15)(12,20)(14,21)(16,23)(18,22)(19,24)(25,30)(26,33)(27,37)(28,34)(29,36)(31,39)(32,40)(35,38)Polycyclic presentation:
2^9|a^2=hi,b^2=gi,c^2=h,d^2=h,e^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^c=egi,f^d=fgi83 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8429351);