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 | |
5169 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 32 points:
(9,10,13,11)(12,15,16,14)(17,18)(19,20)(21,29)(22,23)(24,31)(25,28)(26,30)(27,32);(1,2)(3,6)(4,5)(7,8)(10,11)(14,15)(17,19,22,25)(18,20,23,28)(21,26,31,32)(24,27,29,30);(2,5)(6,8)(9,11,13,10)(12,15,16,14)(18,23)(19,25)(24,29)(26,32);(3,7)(6,8)(9,12)(10,14)(11,15)(13,16)(17,20)(18,19)(21,30)(22,28)(23,25)(24,32)(26,29)(27,31);(1,3)(2,6)(4,7)(5,8)(17,21,22,31)(18,24,23,29)(19,26,25,32)(20,27,28,30);(1,2)(3,6)(4,5)(7,8)(17,18)(19,20)(21,24)(22,23)(25,28)(26,27)(29,31)(30,32)Polycyclic presentation:
2^9|a^2=g,b^2=h,c^2=g,e^2=h,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,f^c=fgi,e^d=egi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8180654);