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 | |
4514 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2,5,8)(3,6,4,7)(9,10)(11,13)(12,20)(14,15)(16,18)(17,21)(19,22)(23,24)(25,26,31,30)(27,35,36,32)(28,39,38,33)(29,34,37,40);(1,2)(3,6)(4,7)(5,8)(9,11)(10,13)(12,19)(14,18)(15,16)(17,23)(20,22)(21,24)(25,27)(26,32)(28,37)(29,38)(30,35)(31,36)(33,40)(34,39);(1,3,5,4)(2,6,8,7)(10,15)(11,18)(12,21)(22,23)(27,36)(29,37)(32,35)(34,40);(1,3,5,4)(2,6,8,7)(9,11)(10,16)(12,19)(13,15)(14,18)(17,22)(20,23)(21,24)(25,28,31,38)(26,33,30,39)(27,37,36,29)(32,40,35,34);(1,3,5,4)(2,6,8,7)(9,12)(10,17)(11,19)(13,23)(14,21)(15,20)(16,22)(18,24)(25,29,31,37)(26,34,30,40)(27,38,36,28)(32,39,35,33);(1,4,5,3)(2,7,8,6)(9,13,14,16)(10,11,15,18)(12,22,21,23)(17,24,20,19)(25,30,31,26)(27,32,36,35)(28,39,38,33)(29,34,37,40)Polycyclic presentation:
2^9|a^2=g,c^2=ghi,d^2=g,e^2=g,f^2=gh,b^a=bg,c^a=ch,d^a=di,e^a=eh,c^b=ci,f^b=fi,d^c=dh,e^c=ei,f^e=fi101 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7880171);