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 | |
4344 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(2,4)(6,8)(9,10)(11,16)(12,17)(13,18)(14,15)(19,24)(20,23)(21,22)(27,33)(31,38)(34,36)(39,40);(3,7)(6,8)(10,15)(12,22)(16,24)(18,20)(25,26)(27,31)(28,35)(29,30)(32,37)(33,38)(34,40)(36,39);(1,2,5,4)(3,6,7,8)(11,19)(13,23)(16,24)(18,20)(26,30)(27,33)(32,35)(34,36);(9,11,14,19)(10,16,15,24)(12,18,22,20)(13,21,23,17)(25,27)(26,31)(28,36)(29,33)(30,38)(32,40)(34,37)(35,39);(1,3)(2,6)(4,8)(5,7)(9,12,14,22)(10,17,15,21)(11,20,19,18)(13,16,23,24)(25,28,29,37)(26,32,30,35)(27,34,33,36)(31,39,38,40);(1,4,5,2)(3,8,7,6)(9,13,14,23)(10,18,15,20)(11,21,19,17)(12,16,22,24)(25,26)(27,31)(28,32)(29,30)(33,38)(34,39)(35,37)(36,40)Polycyclic presentation:
2^9|c^2=h,d^2=g,e^2=gi,f^2=gh,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eghi,d^c=dgi,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, 8838596);