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 | |
4534 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(2,3)(6,7)(9,10)(11,15)(12,17)(13,21)(14,16)(18,22)(19,20)(23,24)(25,26,31,30)(27,34,32,28)(29,37,38,33)(35,36,39,40);(1,2,5,3)(4,6,8,7)(9,11,14,17)(10,15,16,12)(13,19,22,24)(18,23,21,20)(25,27,31,32)(26,28,30,34)(29,35,38,39)(33,36,37,40);(1,2,5,3)(4,7,8,6)(10,16)(11,17)(18,21)(19,24)(25,28,31,34)(26,32,30,27)(29,36,38,40)(33,39,37,35);(1,2,5,3)(4,6,8,7)(9,12,14,15)(10,17,16,11)(13,20,22,23)(18,24,21,19)(25,27)(26,28)(29,35)(30,34)(31,32)(33,36)(37,40)(38,39);(1,3,5,2)(4,7,8,6)(9,13,14,22)(10,18,16,21)(11,19,17,24)(12,20,15,23)(25,29,31,38)(26,33,30,37)(27,35,32,39)(28,36,34,40);(1,4,5,8)(2,6,3,7)(9,10)(11,15)(12,17)(13,18)(14,16)(19,23)(20,24)(21,22)(25,30,31,26)(27,34,32,28)(29,37,38,33)(35,40,39,36)Polycyclic presentation:
2^9|a^2=ghi,b^2=h,c^2=g,d^2=gi,e^2=h,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dghi,f^c=fgi92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8094052);