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 | |
4578 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)(3,7)(4,5)(6,8)(9,10)(11,15)(12,20)(13,18)(14,16)(17,22)(19,23)(21,24)(25,26)(27,32)(28,36)(29,33)(30,34)(31,37)(35,39)(38,40);(2,5)(6,7)(9,11,14,18)(10,15,16,13)(12,19,22,24)(17,21,20,23)(25,27)(26,29)(28,35)(30,33)(31,39)(32,34)(36,38)(37,40);(1,2)(3,6)(4,5)(7,8)(10,16)(11,18)(17,20)(19,24)(28,37)(31,36)(35,40)(38,39);(1,3)(2,6)(4,8)(5,7)(9,11)(10,15)(12,19)(13,16)(14,18)(17,21)(20,23)(22,24)(25,28,30,37)(26,31,34,36)(27,35,33,40)(29,39,32,38);(9,12,14,22)(10,17,16,20)(11,19,18,24)(13,23,15,21)(25,27)(26,32)(28,35)(29,34)(30,33)(31,38)(36,39)(37,40);(9,13,14,15)(10,18,16,11)(12,21,22,23)(17,19,20,24)(25,29,30,32)(26,33,34,27)(28,38,37,39)(31,35,36,40)Polycyclic presentation:
2^9|b^2=h,d^2=ghi,e^2=h,f^2=gi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,d^c=dgi,f^d=fgi,f^e=fgi89 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8632015);