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 | |
4890 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 32 points:
(1,2,4,6)(3,5,7,8)(10,14)(12,16)(17,18,22,23)(19,25,26,20)(21,29,30,24)(27,28,32,31);(3,7)(5,8)(10,14)(11,15)(18,23)(20,25)(24,29)(28,31);(1,2)(3,5)(4,6)(7,8)(9,10)(11,12)(13,14)(15,16)(18,23)(19,26)(21,30)(28,31);(10,14)(11,15)(17,19,22,26)(18,20,23,25)(21,27,30,32)(24,28,29,31);(1,3,4,7)(2,5,6,8)(9,11,13,15)(10,12,14,16)(17,20)(18,19)(21,28)(22,25)(23,26)(24,27)(29,32)(30,31);(1,2)(3,5)(4,6)(7,8)(9,12)(10,15)(11,14)(13,16)(17,21)(18,24)(19,27)(20,28)(22,30)(23,29)(25,31)(26,32)Polycyclic presentation:
2^9|a^2=hi,d^2=g,e^2=gh,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dgi,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, 8812572);