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 | |
4983 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 32 points:
(9,10,13,15)(11,12,16,14)(17,18,22,23)(19,20,27,24)(21,25,31,32)(26,28,30,29);(1,2)(3,6)(4,7)(5,8)(9,11,13,16)(10,14,15,12)(18,23)(20,24)(25,32)(28,29);(1,3)(2,4)(5,7)(6,8)(9,11)(10,12)(13,16)(14,15)(17,19)(18,24)(20,23)(21,26)(22,27)(25,29)(28,32)(30,31);(1,3)(2,6)(4,8)(5,7)(9,12)(10,11)(13,14)(15,16)(17,20,22,24)(18,19,23,27)(21,29,31,28)(25,30,32,26);(1,3)(2,6)(4,8)(5,7)(17,19)(18,24)(20,23)(21,30)(22,27)(25,28)(26,31)(29,32);(1,4,5,6)(2,7,8,3)(17,21,22,31)(18,25,23,32)(19,26,27,30)(20,28,24,29)Polycyclic presentation:
2^9|a^2=g,b^2=gh,d^2=h,f^2=gi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,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, 8584160);