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 |
| prime radical | #59029 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4962 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 32 points:
(11,15)(13,16)(17,18,23,27)(19,28,30,24)(20,29,32,25)(21,22,26,31);(1,2)(3,7)(4,6)(5,8)(9,10,12,14)(11,13,15,16)(17,19,23,30)(18,24,27,28)(20,31,32,22)(21,29,26,25);(1,3)(2,5)(4,8)(6,7)(9,11)(10,13)(12,15)(14,16);(2,6)(5,7)(9,10)(11,13)(12,14)(15,16)(17,20,23,32)(18,25,27,29)(19,22,30,31)(21,24,26,28);(1,3)(2,7)(4,8)(5,6)(9,11)(10,13)(12,15)(14,16)(17,21)(18,22)(19,29)(20,24)(23,26)(25,30)(27,31)(28,32);(1,2)(3,5)(4,6)(7,8)(17,22)(18,26)(19,20)(21,27)(23,31)(24,25)(28,29)(30,32)Polycyclic presentation:
2^9|a^2=g,b^2=gh,d^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dghi,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, 8596184);