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:
(1,2,5,8)(3,6,4,7)(9,10)(11,21)(12,17)(13,18)(14,19)(15,20)(16,22)(23,24)(27,29)(32,34)(35,36)(39,40);(1,2)(3,6)(4,7)(5,8)(9,11,15,22)(10,16,20,21)(12,14,18,23)(13,19,17,24)(25,26)(27,34)(28,33)(29,32)(30,31)(35,40)(36,39)(37,38);(11,22)(14,23)(16,21)(19,24)(25,27,31,29)(26,32,30,34)(28,36,38,35)(33,40,37,39);(9,12)(10,17)(11,14)(13,20)(15,18)(16,24)(19,21)(22,23)(25,28,31,38)(26,33,30,37)(27,35,29,36)(32,39,34,40);(1,3,5,4)(2,6,8,7)(9,13,15,17)(10,18,20,12)(11,19,22,24)(14,16,23,21)(25,29,31,27)(26,32,30,34)(28,35,38,36)(33,40,37,39);(1,4,5,3)(2,7,8,6)(9,14)(10,19)(11,12)(13,16)(15,23)(17,21)(18,22)(20,24)(25,30,31,26)(27,32,29,34)(28,37,38,33)(35,39,36,40)Polycyclic presentation:
2^9|a^2=ghi,b^2=hi,c^2=h,d^2=h,e^2=gh,f^2=gi,b^a=bg,c^a=ch,e^a=eh,c^b=ci,f^b=fi,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, 7749642);