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 | |
4387 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2,5,7)(3,4,8,6)(9,10)(11,12)(13,17)(14,22)(15,16)(18,23)(19,24)(20,21);(1,3,5,8)(2,6,7,4)(9,11)(10,12)(13,15)(14,19)(16,17)(18,20)(21,23)(22,24)(25,26,30,35)(27,33,36,31)(28,32,38,39)(29,34,40,37);(1,2)(3,6)(4,8)(5,7)(10,16)(11,13)(14,23)(20,24)(25,27,30,36)(26,31,35,33)(28,34,38,37)(29,39,40,32);(1,3,5,8)(2,6,7,4)(9,12)(10,11)(13,16)(14,20)(15,17)(18,19)(21,22)(23,24)(25,28)(26,32)(27,34)(29,31)(30,38)(33,40)(35,39)(36,37);(1,3)(2,4)(5,8)(6,7)(9,13,15,11)(10,17,16,12)(14,21,23,19)(18,24,22,20)(25,27)(26,33)(28,37)(29,32)(30,36)(31,35)(34,38)(39,40);(1,4)(2,3)(5,6)(7,8)(9,14,15,23)(10,18,16,22)(11,19,13,21)(12,20,17,24)(25,29)(26,34)(27,32)(28,31)(30,40)(33,38)(35,37)(36,39)Polycyclic presentation:
2^9|a^2=g,b^2=hi,c^2=ghi,d^2=g,e^2=gh,f^2=gh,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,e^c=egi,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, 8853420);