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 | |
4164 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)(11,16)(12,13)(14,23)(15,17)(18,20)(19,24)(21,22)(26,32)(28,36)(33,38)(34,37);(2,3)(6,7)(9,11,15,19)(10,16,17,24)(12,20,21,14)(13,18,22,23)(25,26)(27,33)(28,30)(29,38)(31,32)(34,35)(36,40)(37,39);(1,3,5,2)(4,7,8,6)(10,17)(11,19)(12,21)(18,23)(25,27,31,29)(26,33,32,38)(28,34,36,37)(30,35,40,39);(1,4,5,8)(2,6,3,7)(9,12)(10,13)(11,20)(14,19)(15,21)(16,18)(17,22)(23,24)(25,28)(26,30)(27,34)(29,37)(31,36)(32,40)(33,35)(38,39);(9,13)(10,12)(11,18)(14,24)(15,22)(16,20)(17,21)(19,23)(25,29,31,27)(26,33,32,38)(28,37,36,34)(30,35,40,39);(9,14,15,20)(10,18,17,23)(11,12,19,21)(13,16,22,24)(25,30)(26,28)(27,35)(29,39)(31,40)(32,36)(33,34)(37,38)Polycyclic presentation:
2^9|a^2=hi,b^2=h,c^2=g,d^2=hi,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, 9770461);