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 | #49863 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4324 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)(4,5)(6,8)(9,10)(11,15)(12,16)(13,17)(14,18)(19,22)(20,23)(21,24)(25,26,31,35)(27,32,38,39)(28,30,37,33)(29,34,36,40);(2,5)(6,7)(9,10,14,18)(11,15,16,12)(13,17,21,24)(19,22,23,20)(25,27,31,38)(26,32,35,39)(28,29,37,36)(30,34,33,40);(1,2)(3,6)(4,5)(7,8)(11,16)(12,15)(13,21)(17,24)(25,28,31,37)(26,33,35,30)(27,36,38,29)(32,34,39,40);(1,3)(2,6)(4,8)(5,7)(9,11)(10,15)(12,18)(13,19)(14,16)(17,22)(20,24)(21,23)(25,28)(26,33)(27,29)(30,35)(31,37)(32,40)(34,39)(36,38);(9,12,14,15)(10,16,18,11)(13,20,21,22)(17,23,24,19)(25,29)(26,34)(27,37)(28,38)(30,39)(31,36)(32,33)(35,40);(9,13)(10,17)(11,19)(12,20)(14,21)(15,22)(16,23)(18,24)(25,30,31,33)(26,28,35,37)(27,34,38,40)(29,39,36,32)Polycyclic presentation:
2^9|a^2=h,b^2=gi,c^2=h,e^2=ghi,f^2=h,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eghi,d^c=dgi,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, 8961595);