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 | #43105 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4408 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)(9,10,15,17)(11,16,12,22)(13,18,19,14)(20,21,23,24)(25,26)(27,31)(28,34)(29,32)(30,33)(35,39)(36,40)(37,38);(2,3)(6,7)(9,11)(10,16)(12,15)(13,23)(14,21)(17,22)(18,24)(19,20)(27,28)(31,34)(35,36)(39,40);(2,3)(6,7)(10,17)(11,12)(13,19)(21,24)(25,27,30,28)(26,31,33,34)(29,35,38,36)(32,39,37,40);(1,2,5,3)(4,6,8,7)(9,12,15,11)(10,16,17,22)(13,23,19,20)(14,24,18,21)(25,28,30,27)(26,31,33,34)(29,36,38,35)(32,39,37,40);(1,3,5,2)(4,6,8,7)(9,13)(10,18)(11,20)(12,23)(14,17)(15,19)(16,21)(22,24)(25,26)(27,34)(28,31)(29,37)(30,33)(32,38)(35,39)(36,40);(1,4)(2,6)(3,7)(5,8)(9,14,15,18)(10,19,17,13)(11,21,12,24)(16,20,22,23)(25,29)(26,32)(27,35)(28,36)(30,38)(31,39)(33,37)(34,40)Polycyclic presentation:
2^9|a^2=h,c^2=ghi,d^2=gi,e^2=g,f^2=gh,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,d^b=dgi,e^b=eh,d^c=dh,e^c=egi,f^e=fgi83 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 9954658);