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 | #39448 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4249 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)(11,16)(12,21)(13,18)(14,22)(15,17)(19,24)(20,23)(25,26,30,34)(27,32,37,31)(28,29,33,39)(35,38,40,36);(2,3)(6,7)(9,11,15,12)(10,16,17,21)(13,22,23,19)(14,20,24,18)(25,27)(26,31)(28,35)(29,36)(30,37)(32,34)(33,40)(38,39);(1,3,5,2)(4,7,8,6)(10,17)(11,12)(18,20)(19,22);(1,4,5,8)(2,6,3,7)(9,12,15,11)(10,16,17,21)(13,22,23,19)(14,18,24,20)(25,28,30,33)(26,29,34,39)(27,35,37,40)(31,36,32,38);(9,13,15,23)(10,18,17,20)(11,19,12,22)(14,16,24,21)(25,27)(26,32)(28,35)(29,38)(30,37)(31,34)(33,40)(36,39);(9,14,15,24)(10,19,17,22)(11,20,12,18)(13,21,23,16)(25,29)(26,33)(27,36)(28,34)(30,39)(31,40)(32,35)(37,38)Polycyclic presentation:
2^9|a^2=g,b^2=h,c^2=hi,d^2=gh,e^2=h,f^2=h,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eh,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, 9633622);