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 | |
4508 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)(3,8)(4,7)(5,6)(9,10,14,17)(11,15,18,22)(12,13,16,21)(19,20,23,24)(25,26)(27,32)(28,30)(29,34)(31,33)(35,37)(36,38)(39,40);(1,2,6,5)(3,7,4,8)(9,11)(10,15)(12,19)(13,20)(14,18)(16,23)(17,22)(21,24)(25,27,31,36)(26,32,33,38)(28,34,35,40)(29,37,39,30);(1,3,6,4)(2,7,5,8)(11,18)(15,22)(19,23)(20,24)(26,33)(27,36)(30,37)(34,40);(1,2,6,5)(3,7,4,8)(9,12)(10,13)(11,19)(14,16)(15,20)(17,21)(18,23)(22,24)(25,28,31,35)(26,30,33,37)(27,34,36,40)(29,38,39,32);(1,4,6,3)(2,7,5,8)(9,13)(10,16)(11,20)(12,17)(14,21)(15,23)(18,24)(19,22)(25,29)(26,34)(27,30)(28,32)(31,39)(33,40)(35,38)(36,37);(1,5,6,2)(3,7,4,8)(9,11)(10,15)(12,19)(13,20)(14,18)(16,23)(17,22)(21,24)(25,30)(26,35)(27,29)(28,33)(31,37)(32,40)(34,38)(36,39)Polycyclic presentation:
2^9|a^2=ghi,b^2=h,c^2=g,d^2=h,e^2=g,f^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,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, 8800933);