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 | |
4393 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 40 points:
(9,10)(11,17)(12,13)(14,15)(16,20)(18,23)(19,24)(21,22)(25,26,31,33)(27,35,29,32)(28,38,34,30)(36,37,39,40);(2,3)(6,7)(10,15)(12,21)(16,17)(18,24)(25,27,31,29)(26,32,33,35)(28,36,34,39)(30,37,38,40);(1,2,5,3)(4,6,8,7)(26,33)(28,34)(32,35)(36,39);(9,11,14,20)(10,16,15,17)(12,18,21,24)(13,19,22,23)(25,28,31,34)(26,30,33,38)(27,36,29,39)(32,37,35,40);(1,3,5,2)(4,6,8,7)(9,12,14,21)(10,13,15,22)(11,18,20,24)(16,19,17,23)(25,29,31,27)(26,32,33,35)(28,39,34,36)(30,37,38,40);(1,4)(2,6)(3,7)(5,8)(9,13)(10,12)(11,19)(14,22)(15,21)(16,18)(17,24)(20,23)(25,30)(26,34)(27,37)(28,33)(29,40)(31,38)(32,39)(35,36)Polycyclic presentation:
2^9|a^2=h,b^2=h,c^2=i,d^2=g,e^2=gi,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eghi,d^c=dh,f^e=fgi83 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 9755019);