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 | |
4630 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2,5,7)(3,4,8,6)(9,10)(11,13)(12,16)(14,18)(15,20)(17,22)(19,23)(21,24)(25,26,32,36)(27,29,30,33)(28,35,31,34)(37,38,40,39);(1,3,5,8)(2,6,7,4)(9,11,14,20)(10,15,18,13)(12,19,22,24)(16,21,17,23)(25,27)(26,33)(28,37)(29,36)(30,32)(31,40)(34,38)(35,39);(1,3)(2,4)(5,8)(6,7)(9,12,14,22)(10,16,18,17)(11,19,20,24)(13,23,15,21)(25,28,32,31)(26,34,36,35)(27,37,30,40)(29,39,33,38);(1,4)(2,3)(5,6)(7,8)(9,12)(10,17)(11,19)(13,21)(14,22)(15,23)(16,18)(20,24)(25,29,32,33)(26,27,36,30)(28,38,31,39)(34,40,35,37);(9,13)(10,11)(12,21)(14,15)(16,24)(17,19)(18,20)(22,23)(25,30,32,27)(26,33,36,29)(28,37,31,40)(34,39,35,38);(9,11)(10,13)(12,19)(14,20)(15,18)(16,23)(17,21)(22,24)(25,31,32,28)(26,35,36,34)(27,37,30,40)(29,39,33,38)Polycyclic presentation:
2^9|a^2=hi,b^2=gh,c^2=gi,d^2=h,e^2=h,f^2=h,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,f^b=fgi,d^c=dgi,e^c=egi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8679963);