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 | |
4386 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 40 points:
(2,5)(6,8)(25,26,31,30)(27,35,29,32)(28,34,38,33)(36,39,37,40);(1,2)(3,6)(4,5)(7,8)(9,10,14,15)(11,16,20,22)(12,19,23,17)(13,18,24,21)(25,27)(26,32)(28,37)(29,31)(30,35)(33,40)(34,39)(36,38);(10,15)(13,24)(16,22)(17,19)(25,28,31,38)(26,33,30,34)(27,36,29,37)(32,39,35,40);(1,3,4,7)(2,6,5,8)(9,11)(10,16)(12,21)(13,19)(14,20)(15,22)(17,24)(18,23)(25,28)(26,34)(27,37)(29,36)(30,33)(31,38)(32,39)(35,40);(9,12,14,23)(10,17,15,19)(11,18,20,21)(13,22,24,16)(25,29,31,27)(26,35,30,32)(28,37,38,36)(33,40,34,39);(9,13)(10,18)(11,19)(12,22)(14,24)(15,21)(16,23)(17,20)(25,30,31,26)(27,35,29,32)(28,33,38,34)(36,39,37,40)Polycyclic presentation:
2^9|a^2=h,b^2=hi,c^2=h,d^2=gh,e^2=i,f^2=h,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, 7702909);