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 | |
4500 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(2,5)(6,8)(9,10)(11,15)(12,13)(14,17)(16,20)(18,19)(21,24)(22,23)(25,26,30,33)(27,34,29,31)(28,32,37,40)(35,39,36,38);(3,7)(6,8)(9,11)(10,15)(12,18)(13,19)(14,20)(16,17)(21,22)(23,24)(25,26)(27,31)(28,32)(29,34)(30,33)(35,38)(36,39)(37,40);(1,2)(3,6)(4,5)(7,8)(9,11,14,20)(10,15,17,16)(12,18,22,21)(13,19,23,24)(25,27,30,29)(26,31,33,34)(28,36,37,35)(32,39,40,38);(9,11)(10,16)(12,21)(13,19)(14,20)(15,17)(18,22)(23,24)(25,28,30,37)(26,32,33,40)(27,35,29,36)(31,38,34,39);(1,3)(2,6)(4,7)(5,8)(9,12,14,22)(10,13,17,23)(11,18,20,21)(15,19,16,24)(25,29,30,27)(26,31,33,34)(28,35,37,36)(32,39,40,38);(1,2)(3,6)(4,5)(7,8)(9,13)(10,12)(11,19)(14,23)(15,18)(16,21)(17,22)(20,24)(25,26)(27,31)(28,32)(29,34)(30,33)(35,38)(36,39)(37,40)Polycyclic presentation:
2^9|a^2=g,c^2=gi,d^2=g,e^2=gi,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eh,d^c=dgi,e^d=egi,f^e=fgi83 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 9113856);