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 | |
4495 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2,5,3)(4,7,8,6)(10,11)(15,17)(16,18)(22,23)(25,26,31,30)(27,32,34,28)(29,37,39,33)(35,36,40,38);(2,3)(6,7)(9,10,14,11)(12,15,20,17)(13,16,21,18)(19,22,24,23)(25,27)(26,32)(28,30)(29,35)(31,34)(33,38)(36,37)(39,40);(1,3,5,2)(4,7,8,6)(13,21)(16,18)(19,24)(22,23)(26,30)(27,34)(33,37)(35,40);(1,4)(2,6)(3,7)(5,8)(9,11,14,10)(12,15,20,17)(13,18,21,16)(19,22,24,23)(25,28)(26,27)(29,38)(30,34)(31,32)(33,40)(35,37)(36,39);(9,12)(10,15)(11,17)(13,19)(14,20)(16,22)(18,23)(21,24)(25,29,31,39)(26,33,30,37)(27,35,34,40)(28,36,32,38);(9,13,14,21)(10,16,11,18)(12,19,20,24)(15,22,17,23)(25,30,31,26)(27,28,34,32)(29,37,39,33)(35,36,40,38)Polycyclic presentation:
2^9|a^2=i,b^2=ghi,c^2=hi,d^2=ghi,e^2=h,f^2=gi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,f^c=fgi,e^d=egi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8247089);