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 | |
4732 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2,4,6)(3,5,7,8)(9,10)(11,15)(12,13)(14,17)(16,20)(18,19)(21,24)(22,23)(25,26,31,29)(27,35,30,32)(28,34,38,33)(36,39,37,40);(3,7)(5,8)(9,11)(10,15)(12,18)(13,19)(14,20)(16,17)(21,22)(23,24)(25,27,31,30)(26,32,29,35)(28,36,38,37)(33,39,34,40);(1,2)(3,5)(4,6)(7,8)(9,11,14,20)(10,15,17,16)(12,18,22,21)(13,19,23,24)(25,28,31,38)(26,33,29,34)(27,36,30,37)(32,39,35,40);(9,11)(10,16)(12,21)(13,19)(14,20)(15,17)(18,22)(23,24)(25,28)(26,34)(27,36)(29,33)(30,37)(31,38)(32,40)(35,39);(1,3)(2,5)(4,7)(6,8)(9,12,14,22)(10,13,17,23)(11,18,20,21)(15,19,16,24)(25,29,31,26)(27,32,30,35)(28,34,38,33)(36,39,37,40);(1,2)(3,5)(4,6)(7,8)(9,13)(10,12)(11,19)(14,23)(15,18)(16,21)(17,22)(20,24)(25,30,31,27)(26,35,29,32)(28,37,38,36)(33,40,34,39)Polycyclic presentation:
2^9|a^2=h,b^2=g,c^2=gi,e^2=gi,f^2=g,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eh,d^c=dgi,e^d=egi,f^e=fgi83 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 9113883);