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 | |
4384 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2,5,4)(3,7,8,6)(9,10,14,11)(12,15,20,17)(13,18,22,16)(19,24,21,23)(25,26)(27,30)(28,31)(29,34)(32,38)(33,36)(35,40)(37,39);(1,3,5,8)(2,6,4,7)(10,11)(15,17)(16,18)(23,24)(25,26)(27,30)(28,31)(29,32)(33,36)(34,38)(35,39)(37,40);(10,11)(15,17)(16,18)(23,24)(26,27)(31,33)(32,34)(39,40);(1,2,5,4)(3,7,8,6)(9,11,14,10)(12,17,20,15)(13,18,22,16)(19,24,21,23)(25,27,30,26)(28,33,36,31)(29,34,38,32)(35,40,37,39);(1,3)(2,7)(4,6)(5,8)(9,12,14,20)(10,15,11,17)(13,21,22,19)(16,24,18,23)(25,28,30,36)(26,31,27,33)(29,37,38,35)(32,40,34,39);(1,4,5,2)(3,7,8,6)(9,13,14,22)(10,16,11,18)(12,19,20,21)(15,23,17,24)(25,29)(26,32)(27,34)(28,35)(30,38)(31,39)(33,40)(36,37)Polycyclic presentation:
2^9|a^2=g,b^2=ghi,d^2=gi,e^2=h,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, 9854585);