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 | |
4671 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 32 points:
(9,10)(11,12)(13,14)(15,16)(17,18)(19,27)(20,21)(22,23)(24,30)(25,26)(28,29)(31,32);(1,2,5,3)(4,7,8,6)(10,14)(11,15)(18,23)(21,25)(24,27)(29,32);(2,3)(6,7)(9,11,13,15)(10,12,14,16)(17,19,22,30)(18,24,23,27)(20,31,26,28)(21,29,25,32);(1,2,5,3)(4,6,8,7)(9,12)(10,15)(11,14)(13,16)(17,20,22,26)(18,25,23,21)(19,28,30,31)(24,29,27,32);(1,3,5,2)(4,7,8,6)(17,18)(19,27)(20,25)(21,26)(22,23)(24,30)(28,32)(29,31);(1,4,5,8)(2,6,3,7)(17,21,22,25)(18,26,23,20)(19,29,30,32)(24,28,27,31)Polycyclic presentation:
2^9|b^2=ghi,c^2=g,d^2=gi,e^2=ghi,f^2=gi,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,f^b=fgi,d^c=dgi,e^c=egi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8730995);