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 | |
5157 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)(17,18,22,24)(19,23,27,30)(20,21,28,29)(25,26,31,32);(1,2,5,6)(3,7,8,4)(10,11)(14,15)(17,19,22,27)(18,23,24,30)(20,25,28,31)(21,26,29,32);(1,3)(2,4)(5,8)(6,7)(9,11,13,10)(12,15,16,14)(18,24)(20,28)(23,30)(25,31);(2,6)(4,7)(9,12)(10,14)(11,15)(13,16)(17,20)(18,21)(19,25)(22,28)(23,26)(24,29)(27,31)(30,32);(1,4)(2,3)(5,7)(6,8)(17,19)(18,23)(20,25)(21,26)(22,27)(24,30)(28,31)(29,32);(1,3)(2,7)(4,6)(5,8)(17,21)(18,20)(19,26)(22,29)(23,25)(24,28)(27,32)(30,31)Polycyclic presentation:
2^9|a^2=gh,b^2=ghi,c^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,d^b=dgi,e^b=eh,d^c=dh,e^d=egi,f^e=fgi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 10132770);