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 |
| prime radical | #44130 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4234 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2,5,6)(3,7,8,4)(9,10,14,15)(11,17,19,12)(13,21,23,16)(18,20,22,24)(25,26,31,28)(27,34,30,32)(29,33,37,36)(35,40,38,39);(2,6)(4,7)(9,11,14,19)(10,12,15,17)(13,22,23,18)(16,24,21,20)(29,37)(33,36)(35,38)(39,40);(1,2)(3,4)(5,6)(7,8)(10,15)(12,17)(16,21)(20,24)(25,27,31,30)(26,32,28,34)(29,35,37,38)(33,39,36,40);(1,3)(2,4)(5,8)(6,7)(9,10)(11,12)(13,16)(14,15)(17,19)(18,20)(21,23)(22,24)(25,28,31,26)(27,34,30,32)(29,36,37,33)(35,40,38,39);(1,4)(2,3)(5,7)(6,8)(9,12)(10,11)(13,20)(14,17)(15,19)(16,18)(21,22)(23,24)(25,29,31,37)(26,33,28,36)(27,35,30,38)(32,39,34,40);(1,3)(2,4)(5,8)(6,7)(9,13,14,23)(10,16,15,21)(11,18,19,22)(12,20,17,24)(25,30,31,27)(26,34,28,32)(29,38,37,35)(33,40,36,39)Polycyclic presentation:
2^9|a^2=h,b^2=gi,c^2=gh,d^2=gh,e^2=gh,f^2=hi,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,f^b=fgi,d^c=dgi,e^c=egi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8717411);