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 | |
4361 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 40 points:
(3,4)(6,7)(9,10)(11,19)(12,23)(13,17)(14,18)(15,22)(16,21)(20,24)(28,36)(32,39)(35,37)(38,40);(1,2,5,8)(3,6,4,7)(9,11)(10,15)(12,20)(13,21)(14,22)(16,24)(17,23)(18,19)(25,26)(27,34)(28,32)(29,31)(30,33)(35,40)(36,39)(37,38);(1,3,5,4)(2,6,8,7)(9,10,14,18)(11,15,22,19)(12,16,21,23)(13,17,20,24)(25,27,30,29)(26,31,33,34)(28,37,36,35)(32,40,39,38);(1,2)(3,6)(4,7)(5,8)(9,10)(11,19)(12,16)(13,24)(14,18)(15,22)(17,20)(21,23)(25,28,30,36)(26,32,33,39)(27,35,29,37)(31,38,34,40);(1,4,5,3)(2,7,8,6)(9,12)(10,16)(11,20)(13,22)(14,21)(15,24)(17,19)(18,23)(25,26)(27,31)(28,32)(29,34)(30,33)(35,38)(36,39)(37,40);(9,13,14,20)(10,17,18,24)(11,21,22,12)(15,23,19,16)(25,29,30,27)(26,31,33,34)(28,35,36,37)(32,40,39,38)Polycyclic presentation:
2^9|b^2=gh,c^2=hi,d^2=i,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, 8658326);