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 | |
5234 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 32 points:
(3,7)(5,8)(9,10)(11,12)(13,14)(15,16)(20,28)(24,29)(26,32)(30,31);(1,2,4,6)(3,5,7,8)(10,14)(11,15)(17,18)(19,23)(20,24)(21,22)(25,27)(26,30)(28,29)(31,32);(1,3)(2,5)(4,7)(6,8)(9,11,13,15)(10,12,14,16)(18,22)(23,25)(24,29)(30,31);(1,2)(3,5)(4,6)(7,8)(9,12)(10,15)(11,14)(13,16)(17,18)(19,25)(20,24)(21,22)(23,27)(26,31)(28,29)(30,32);(1,3)(2,5)(4,7)(6,8)(17,19)(18,23)(20,26)(21,27)(22,25)(24,30)(28,32)(29,31);(17,20)(18,24)(19,26)(21,28)(22,29)(23,30)(25,31)(27,32)Polycyclic presentation:
2^9|b^2=h,c^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,f^a=fgi,c^b=ci,d^b=dh,d^c=dgi,e^d=egi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8061899);