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 | |
4609 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2,5,3)(4,7,8,6)(9,10)(11,17)(12,16)(13,18)(14,15)(19,24)(20,21)(22,23)(25,26,29,31)(27,30,33,38)(28,32,35,39)(34,37,36,40);(2,3)(6,7)(9,11)(10,13)(12,20)(14,18)(15,17)(16,24)(19,22)(21,23)(25,27,29,33)(26,30,31,38)(28,34,35,36)(32,37,39,40);(1,3,5,2)(4,7,8,6)(10,15)(11,18)(16,23)(19,20)(26,31)(27,33)(28,35)(37,40);(1,4)(2,6)(3,7)(5,8)(9,11)(10,13)(12,19)(14,18)(15,17)(16,21)(20,22)(23,24)(25,28)(26,32)(27,34)(29,35)(30,37)(31,39)(33,36)(38,40);(9,12)(10,16)(11,19)(13,21)(14,22)(15,23)(17,24)(18,20)(25,27)(26,30)(28,36)(29,33)(31,38)(32,40)(34,35)(37,39);(9,13)(10,11)(12,21)(14,17)(15,18)(16,19)(20,23)(22,24)(25,26)(27,30)(28,32)(29,31)(33,38)(34,37)(35,39)(36,40)Polycyclic presentation:
2^9|a^2=gh,b^2=gi,c^2=hi,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dh,e^c=egi,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, 9401881);