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 | |
4624 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(9,10)(11,17)(12,14)(13,16)(15,18)(19,24)(20,22)(21,23)(25,26)(27,33)(28,30)(29,32)(31,35)(34,40)(36,38)(37,39);(1,2)(3,5)(4,6)(7,8)(10,12)(11,18)(16,20)(19,21)(25,27)(26,31)(28,33)(29,37)(30,35)(32,40)(34,38)(36,39);(2,3)(6,7)(10,12)(15,17)(16,20)(23,24)(29,38)(32,36)(34,37)(39,40);(4,8)(6,7)(9,11,14,18)(10,15,12,17)(13,21,22,19)(16,24,20,23)(25,28,30,26)(27,33,35,31)(29,36,38,32)(34,40,37,39);(1,3,5,2)(4,7,8,6)(9,12,14,10)(11,17,18,15)(13,20,22,16)(19,24,21,23)(25,29)(26,32)(27,34)(28,36)(30,38)(31,39)(33,40)(35,37);(1,4)(2,6)(3,7)(5,8)(9,13,14,22)(10,16,12,20)(11,19,18,21)(15,23,17,24)(25,27)(26,33)(28,31)(29,34)(30,35)(32,40)(36,39)(37,38)Polycyclic presentation:
2^9|d^2=g,e^2=hi,f^2=h,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^b=egi,e^c=egi,f^d=fgi89 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8521373);