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 | |
4643 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)(17,18,23,20)(19,24,27,28)(21,25,31,29)(22,30,32,26);(1,2,5,4)(3,7,8,6)(10,11)(14,15)(17,19)(18,24)(20,28)(21,30)(22,29)(23,27)(25,32)(26,31);(1,3,5,8)(2,6,4,7)(9,11,13,10)(12,15,16,14)(18,20)(19,27)(21,31)(22,32);(3,8)(6,7)(9,12)(10,14)(11,15)(13,16)(17,20,23,18)(19,28,27,24)(21,25,31,29)(22,30,32,26);(1,3)(2,7)(4,6)(5,8)(17,21)(18,25)(19,26)(20,29)(22,24)(23,31)(27,30)(28,32);(1,4,5,2)(3,6,8,7)(17,22,23,32)(18,26,20,30)(19,29,27,25)(21,24,31,28)Polycyclic presentation:
2^9|a^2=gh,b^2=ghi,c^2=hi,d^2=h,f^2=gi,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dgi,e^c=egi,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, 8935615);