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 | |
4373 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 40 points:
(9,10,14,16)(11,19,20,15)(12,21,22,17)(13,18,23,24)(25,26)(27,33)(28,38)(29,35)(30,36)(31,32)(34,39)(37,40);(1,2,5,4)(3,6,7,8)(9,11,14,20)(10,15,16,19)(12,13,22,23)(17,18,21,24)(25,27,32,29)(26,33,31,35)(28,30,39,37)(34,40,38,36);(2,4)(3,7)(10,16)(12,22)(13,23)(15,19)(25,28,32,39)(26,34,31,38)(27,37,29,30)(33,36,35,40);(1,3,5,7)(2,6,4,8)(9,12,14,22)(10,17,16,21)(11,13,20,23)(15,18,19,24)(25,29,32,27)(26,35,31,33)(28,37,39,30)(34,36,38,40);(1,3)(2,6)(4,8)(5,7)(9,13)(10,18)(11,12)(14,23)(15,17)(16,24)(19,21)(20,22)(25,30,32,37)(26,36,31,40)(27,28,29,39)(33,38,35,34);(1,4,5,2)(3,8,7,6)(9,10)(11,15)(12,17)(13,18)(14,16)(19,20)(21,22)(23,24)(25,31,32,26)(27,35,29,33)(28,34,39,38)(30,40,37,36)Polycyclic presentation:
2^9|a^2=g,b^2=gi,c^2=gh,d^2=gi,e^2=gh,f^2=i,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dgi,e^c=egi,f^c=fgi,e^d=egi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8939663);