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 | |
5046 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)(3,8)(4,6)(5,7)(9,10,13,15)(11,12,16,14)(18,19)(21,29)(23,25)(27,30);(1,3,5,4)(2,6,7,8)(9,11,13,16)(10,14,15,12)(18,19)(23,25)(24,26)(31,32);(9,11)(10,12)(13,16)(14,15)(17,18,22,19)(20,23,28,25)(21,24,29,26)(27,31,30,32);(1,2)(3,8)(4,6)(5,7)(9,12)(10,11)(13,14)(15,16)(17,19,22,18)(20,23,28,25)(21,26,29,24)(27,31,30,32);(1,4,5,3)(2,6,7,8)(17,20,22,28)(18,23,19,25)(21,30,29,27)(24,32,26,31);(1,2)(3,6)(4,8)(5,7)(17,21)(18,24)(19,26)(20,27)(22,29)(23,31)(25,32)(28,30)Polycyclic presentation:
2^9|a^2=hi,b^2=g,c^2=h,d^2=h,e^2=gi,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,d^b=dgi,e^d=eh,f^e=fgi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 10412157);