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 | |
4643 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2,5,3)(4,7,8,6)(9,10)(11,12)(13,17)(14,16)(15,19)(18,20)(21,23)(22,24)(25,26,31,32)(27,33,28,29)(30,34,39,40)(35,38,36,37);(2,3)(6,7)(9,11)(10,15)(12,16)(13,18)(14,19)(17,23)(20,24)(21,22)(26,32)(29,33)(34,40)(37,38);(1,3,5,2)(4,7,8,6)(10,16)(12,15)(17,24)(20,23)(25,27,31,28)(26,33,32,29)(30,35,39,36)(34,38,40,37);(1,4,5,8)(2,6,3,7)(9,12,14,15)(10,11,16,19)(13,20,22,23)(17,18,24,21)(25,28,31,27)(26,29,32,33)(30,36,39,35)(34,37,40,38);(9,13,14,22)(10,17,16,24)(11,18,19,21)(12,20,15,23)(25,29)(26,27)(28,32)(30,38)(31,33)(34,36)(35,40)(37,39);(9,11)(10,15)(12,16)(13,21)(14,19)(17,20)(18,22)(23,24)(25,30)(26,34)(27,35)(28,36)(29,37)(31,39)(32,40)(33,38)Polycyclic presentation:
2^9|a^2=gh,c^2=gh,d^2=g,e^2=h,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eghi,d^c=dh,f^e=fgi77 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 9818149);