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 |
| prime radical | #392 | C23 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4154 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2,5,4)(3,7,8,6)(13,21)(17,22)(18,23)(20,24)(25,26)(27,32)(28,29)(30,35)(31,33)(34,40)(36,37)(38,39);(1,3,5,8)(2,6,4,7)(9,10)(11,12)(13,17)(14,15)(16,19)(18,20)(21,22)(23,24)(25,27,31,30)(26,32,33,35)(28,36,34,38)(29,37,40,39);(1,2,5,4)(3,6,8,7)(10,15)(12,19)(17,22)(20,24)(26,33)(28,34)(32,35)(36,38);(1,2,5,4)(3,6,8,7)(9,11,14,16)(10,12,15,19)(13,18,21,23)(17,20,22,24)(25,28,31,34)(26,29,33,40)(27,36,30,38)(32,37,35,39);(1,4,5,2)(3,7,8,6)(9,12)(10,16)(11,15)(13,20)(14,19)(17,23)(18,22)(21,24)(25,29)(26,34)(27,37)(28,33)(30,39)(31,40)(32,38)(35,36);(9,13)(10,17)(11,18)(12,20)(14,21)(15,22)(16,23)(19,24)(25,30,31,27)(26,35,33,32)(28,38,34,36)(29,39,40,37)Polycyclic presentation:
2^9|a^2=g,b^2=gh,c^2=g,d^2=ghi,e^2=g,f^2=h,b^a=bg,c^a=ch,e^a=eh,f^a=fi,c^b=ci,e^b=ei,d^c=dh,e^c=ei98 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7625174);