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 | |
4724 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)(3,7)(4,8)(5,6)(10,11)(15,17)(16,18)(22,23)(25,26,31,35)(27,30,37,32)(28,38,39,33)(29,40,36,34);(1,2,5,6)(3,7,8,4)(9,10,14,11)(12,15,20,17)(13,16,21,18)(19,22,24,23)(25,27,31,37)(26,32,35,30)(28,36,39,29)(33,34,38,40);(1,3)(2,4)(5,8)(6,7)(13,21)(16,18)(19,24)(22,23)(25,28,31,39)(26,33,35,38)(27,36,37,29)(30,40,32,34);(1,4)(2,3)(5,7)(6,8)(9,11,14,10)(12,15,20,17)(13,18,21,16)(19,22,24,23)(25,29)(26,34)(27,28)(30,33)(31,36)(32,38)(35,40)(37,39);(1,2)(3,4)(5,6)(7,8)(9,12)(10,15)(11,17)(13,19)(14,20)(16,22)(18,23)(21,24)(25,27)(26,30)(28,36)(29,39)(31,37)(32,35)(33,40)(34,38);(1,4)(2,3)(5,7)(6,8)(9,13,14,21)(10,16,11,18)(12,19,20,24)(15,22,17,23)(25,30)(26,27)(28,34)(29,33)(31,32)(35,37)(36,38)(39,40)Polycyclic presentation:
2^9|a^2=hi,b^2=g,c^2=hi,d^2=gh,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, 8806135);