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 | |
4337 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)(3,5)(4,7)(6,8)(9,10)(11,15)(12,20)(13,17)(14,18)(16,21)(19,22)(23,24)(25,26,30,27)(28,33,36,31)(29,32,37,34)(35,39,40,38);(2,3)(6,7)(9,11)(10,15)(12,17)(13,20)(14,19)(16,24)(18,22)(21,23)(26,27)(31,33)(32,34)(38,39);(1,2,5,3)(4,6,8,7)(11,19)(12,21)(15,22)(16,20)(25,27,30,26)(28,33,36,31)(29,34,37,32)(35,39,40,38);(1,3,5,2)(4,7,8,6)(9,12,14,21)(10,16,18,20)(11,17,19,23)(13,15,24,22)(25,28)(26,31)(27,33)(29,35)(30,36)(32,38)(34,39)(37,40);(1,4)(2,6)(3,7)(5,8)(9,10)(11,15)(12,16)(13,23)(14,18)(17,24)(19,22)(20,21)(25,29,30,37)(26,32,27,34)(28,35,36,40)(31,38,33,39);(9,13,14,24)(10,17,18,23)(11,20,19,16)(12,22,21,15)Polycyclic presentation:
2^9|a^2=gh,c^2=g,d^2=ghi,e^2=gh,f^2=gi,b^a=bg,c^a=ch,d^a=di,e^a=eh,c^b=ci,d^b=dh,d^c=dgi,f^d=fgi,f^e=fgi83 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8657825);