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 | |
4362 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2,5,3)(4,7,8,6)(9,10,14,16)(11,18,20,15)(12,21,22,17)(13,23,24,19)(26,32)(28,37)(33,39)(35,38);(2,3)(6,7)(9,11)(10,15)(12,19)(13,17)(14,20)(16,18)(21,24)(22,23)(25,26)(27,33)(28,34)(29,39)(30,35)(31,32)(36,37)(38,40);(1,3,5,2)(4,7,8,6)(10,16)(12,22)(15,18)(19,23)(25,27,31,29)(26,33,32,39)(28,35,37,38)(30,36,40,34);(1,4)(2,6)(3,7)(5,8)(9,12,14,22)(10,17,16,21)(11,19,20,23)(13,18,24,15)(25,28,31,37)(26,34,32,36)(27,35,29,38)(30,39,40,33);(9,11)(10,18)(12,19)(13,21)(14,20)(15,16)(17,24)(22,23)(25,29,31,27)(26,33,32,39)(28,38,37,35)(30,36,40,34);(9,13,14,24)(10,19,16,23)(11,17,20,21)(12,18,22,15)(25,30)(26,35)(27,36)(28,39)(29,34)(31,40)(32,38)(33,37)Polycyclic presentation:
2^9|a^2=hi,c^2=gh,d^2=gi,e^2=ghi,f^2=h,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eghi,d^c=dh,f^e=fgi83 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 9770241);