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 | |
4541 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)(9,10,15,13)(11,19,20,16)(12,17,22,23)(14,18,21,24)(25,26)(27,32)(28,30)(29,33)(31,35)(34,40)(36,38)(37,39);(1,3,5,8)(2,6,4,7)(9,11)(10,16)(12,21)(13,19)(14,22)(15,20)(17,18)(23,24)(25,26,30,28)(27,32,35,31)(29,33,38,36)(34,40,37,39);(11,20)(14,21)(16,19)(18,24)(25,27,30,35)(26,31,28,32)(29,37,38,34)(33,40,36,39);(1,2,5,4)(3,6,8,7)(9,12,15,22)(10,17,13,23)(11,21,20,14)(16,18,19,24)(25,27)(26,32)(28,31)(29,34)(30,35)(33,40)(36,39)(37,38);(1,2,5,4)(3,6,8,7)(9,13,15,10)(11,19,20,16)(12,17,22,23)(14,24,21,18)(25,28,30,26)(27,32,35,31)(29,36,38,33)(34,40,37,39);(1,4,5,2)(3,7,8,6)(9,14,15,21)(10,18,13,24)(11,22,20,12)(16,17,19,23)(25,29)(26,33)(27,34)(28,36)(30,38)(31,39)(32,40)(35,37)Polycyclic presentation:
2^9|a^2=gh,b^2=ghi,c^2=h,d^2=gh,e^2=g,f^2=gh,b^a=bg,c^a=ch,e^a=eh,c^b=ci,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, 7675280);