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 | |
4534 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2,4,6)(3,5,7,8)(9,10,14,16)(11,15,18,17)(12,13,22,23)(19,20,21,24)(25,26)(27,35)(28,30)(29,33)(31,34)(32,37)(36,40)(38,39);(3,7)(5,8)(9,11)(10,15)(12,19)(13,20)(14,18)(16,17)(21,22)(23,24)(25,27,31,37)(26,32,34,35)(28,36,38,33)(29,39,40,30);(1,2)(3,5)(4,6)(7,8)(10,16)(11,18)(13,23)(19,21)(25,28,31,38)(26,30,34,39)(27,36,37,33)(29,35,40,32);(9,11)(10,17)(12,21)(13,20)(14,18)(15,16)(19,22)(23,24)(25,28)(26,30)(27,36)(29,32)(31,38)(33,37)(34,39)(35,40);(1,3)(2,5)(4,7)(6,8)(9,12)(10,13)(11,19)(14,22)(15,20)(16,23)(17,24)(18,21)(25,29)(26,33)(27,30)(28,35)(31,40)(32,38)(34,36)(37,39);(1,2)(3,5)(4,6)(7,8)(9,13)(10,12)(11,20)(14,23)(15,19)(16,22)(17,21)(18,24)(25,30)(26,28)(27,29)(31,39)(32,36)(33,35)(34,38)(37,40)Polycyclic presentation:
2^9|a^2=h,b^2=g,c^2=g,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eghi,d^c=dgi,f^c=fgi,e^d=egi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8834048);