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 | |
4467 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2,5,4)(3,7,8,6)(9,10)(11,15)(12,17)(13,18)(14,16)(19,22)(20,23)(21,24)(25,26,31,29)(27,33,35,28)(30,37,38,32)(34,36,40,39);(1,3)(2,6)(4,7)(5,8)(9,11,14,18)(10,15,16,13)(12,20,21,19)(17,23,24,22)(25,27,31,35)(26,28,29,33)(30,34,38,40)(32,36,37,39);(1,2,5,4)(3,6,8,7)(10,16)(11,18)(12,21)(22,23)(26,29)(28,33)(32,37)(36,39);(1,2,5,4)(3,7,8,6)(9,11)(10,13)(12,19)(14,18)(15,16)(17,23)(20,21)(22,24)(25,28)(26,27)(29,35)(30,36)(31,33)(32,34)(37,40)(38,39);(1,3)(2,7)(4,6)(5,8)(9,12,14,21)(10,17,16,24)(11,19,18,20)(13,23,15,22)(25,29,31,26)(27,28,35,33)(30,32,38,37)(34,39,40,36);(1,4,5,2)(3,7,8,6)(9,13)(10,11)(12,22)(14,15)(16,18)(17,20)(19,24)(21,23)(25,30,31,38)(26,32,29,37)(27,34,35,40)(28,36,33,39)Polycyclic presentation:
2^9|a^2=g,b^2=h,c^2=ghi,d^2=ghi,e^2=h,f^2=g,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,d^b=dgi,e^b=eh,d^c=dh,e^c=egi,f^e=fgi83 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 9954195);