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 | #52018 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4142 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)(9,10,15,19)(11,20,21,16)(12,13,17,23)(14,18,22,24)(25,26)(27,35)(28,33)(29,34)(30,31)(32,36)(37,40)(38,39);(1,2)(3,6)(4,7)(5,8)(9,11,15,21)(10,16,19,20)(12,22,17,14)(13,18,23,24)(27,36)(29,37)(32,35)(34,40);(11,21)(14,22)(16,20)(18,24)(25,27,31,36)(26,32,30,35)(28,37,39,29)(33,34,38,40);(9,12)(10,13)(11,22)(14,21)(15,17)(16,18)(19,23)(20,24)(25,28,31,39)(26,33,30,38)(27,29,36,37)(32,40,35,34);(1,3,5,4)(2,6,8,7)(9,13)(10,17)(11,18)(12,19)(14,16)(15,23)(20,22)(21,24)(25,29)(26,34)(27,28)(30,40)(31,37)(32,38)(33,35)(36,39);(1,4,5,3)(2,7,8,6)(9,14,15,22)(10,18,19,24)(11,12,21,17)(13,16,23,20)(25,30,31,26)(27,32,36,35)(28,38,39,33)(29,40,37,34)Polycyclic presentation:
2^9|a^2=g,b^2=hi,c^2=h,d^2=h,e^2=ghi,f^2=gh,b^a=bg,c^a=ch,f^a=fh,c^b=ci,e^b=eh,d^c=dh,f^c=fi,e^d=ei92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7919197);