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 | |
4370 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2,5,8)(3,6,4,7)(9,10)(11,19)(12,17)(13,23)(14,16)(15,20)(18,21)(22,24);(1,2)(3,6)(4,7)(5,8)(9,11)(10,15)(12,21)(13,22)(14,20)(16,19)(17,23)(18,24)(25,26,30,31)(27,32,36,38)(28,35,39,33)(29,34,40,37);(1,3,5,4)(2,6,8,7)(10,16)(11,20)(12,22)(18,23)(26,31)(29,40)(32,38)(33,35);(9,12,14,22)(10,17,16,24)(11,21,20,13)(15,23,19,18)(25,27)(26,32)(28,37)(29,35)(30,36)(31,38)(33,40)(34,39);(9,13,14,21)(10,18,16,23)(11,12,20,22)(15,17,19,24)(25,28,30,39)(26,33,31,35)(27,34,36,37)(29,38,40,32);(1,4,5,3)(2,7,8,6)(9,10)(11,15)(12,17)(13,18)(14,16)(19,20)(21,23)(22,24)(25,29)(26,34)(27,35)(28,38)(30,40)(31,37)(32,39)(33,36)Polycyclic presentation:
2^9|a^2=gh,b^2=hi,c^2=gh,d^2=h,e^2=i,f^2=gh,b^a=bg,c^a=ch,e^a=eh,c^b=ci,e^b=ei,d^c=dh,f^c=fi,e^d=ei98 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7702938);