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 | #45105 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4344 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2,6,8)(3,5,7,4)(9,10)(11,16)(12,17)(13,18)(14,15)(19,21)(20,23)(22,24)(25,26,31,35)(27,36,38,29)(28,33,40,32)(30,39,37,34);(1,3)(2,5)(4,8)(6,7)(10,15)(11,18)(12,20)(21,22)(25,27)(26,29)(28,30)(31,38)(32,39)(33,34)(35,36)(37,40);(1,4,6,5)(2,7,8,3)(25,28,31,40)(26,32,35,33)(27,30,38,37)(29,39,36,34);(1,2)(3,4)(5,7)(6,8)(9,11,14,18)(10,16,15,13)(12,19,20,24)(17,21,23,22)(25,28)(26,33)(27,37)(29,39)(30,38)(31,40)(32,35)(34,36);(1,5,6,4)(2,7,8,3)(9,12,14,20)(10,17,15,23)(11,19,18,24)(13,22,16,21)(25,29)(26,27)(28,39)(30,32)(31,36)(33,37)(34,40)(35,38);(1,2)(3,5)(4,7)(6,8)(9,13,14,16)(10,18,15,11)(12,21,20,22)(17,19,23,24)(25,30,31,37)(26,34,35,39)(27,28,38,40)(29,33,36,32)Polycyclic presentation:
2^9|a^2=h,c^2=h,d^2=ghi,e^2=gh,f^2=g,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,d^b=dgi,e^b=eghi,d^c=dh,f^e=fgi83 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 9852761);