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 | |
4371 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2,5,6)(3,7,8,4)(9,10,14,15)(11,19,21,16)(12,17,22,24)(13,20,23,18)(25,26)(27,29)(28,33)(30,39)(31,32)(34,38)(35,36)(37,40);(2,6)(4,7)(10,15)(12,22)(16,19)(18,20)(25,27,31,30)(26,29,32,39)(28,34,37,36)(33,38,40,35);(1,2)(3,4)(5,6)(7,8)(10,15)(13,23)(16,19)(17,24)(26,32)(27,30)(33,40)(34,36);(1,3)(2,4)(5,8)(6,7)(9,11,14,21)(10,16,15,19)(12,18,22,20)(13,17,23,24)(25,28,31,37)(26,33,32,40)(27,34,30,36)(29,38,39,35);(1,4)(2,3)(5,7)(6,8)(9,12,14,22)(10,17,15,24)(11,20,21,18)(13,19,23,16)(25,29)(26,27)(28,35)(30,32)(31,39)(33,36)(34,40)(37,38);(1,3)(2,4)(5,8)(6,7)(9,13)(10,18)(11,17)(12,19)(14,23)(15,20)(16,22)(21,24)(25,30,31,27)(26,29,32,39)(28,36,37,34)(33,38,40,35)Polycyclic presentation:
2^9|a^2=g,b^2=gh,d^2=gi,e^2=hi,f^2=gh,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,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, 8808832);