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 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4311 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2,5,3)(4,7,8,6)(9,10,14,11)(12,15,19,17)(13,18,21,16)(20,23,24,22)(25,26)(27,30)(28,36)(29,34)(31,32)(33,38)(35,40)(37,39);(2,3)(6,7)(12,19)(15,17)(20,24)(22,23)(25,27)(26,30)(28,37)(29,31)(32,34)(33,40)(35,38)(36,39);(1,3,5,2)(4,7,8,6)(10,11)(15,17)(16,18)(22,23)(26,32)(27,29)(28,38)(39,40);(1,4)(2,6)(3,7)(5,8)(9,11,14,10)(12,17,19,15)(13,18,21,16)(20,23,24,22)(25,28)(26,33)(27,35)(29,37)(30,39)(31,38)(32,36)(34,40);(9,12,14,19)(10,15,11,17)(13,20,21,24)(16,22,18,23)(25,29,31,27)(26,34,32,30)(28,37,38,35)(33,40,36,39);(9,13,14,21)(10,16,11,18)(12,20,19,24)(15,22,17,23)(25,30,31,34)(26,29,32,27)(28,39,38,40)(33,37,36,35)Polycyclic presentation:
2^9|a^2=hi,c^2=g,d^2=ghi,e^2=h,f^2=h,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, 9948045);