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 | |
4646 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)(3,7)(4,8)(5,6)(17,18,23,24)(19,29,31,25)(20,30,32,26)(21,27,28,22);(1,3)(2,4)(5,8)(6,7)(9,10)(11,15)(12,14)(13,16)(18,24)(20,32)(21,28)(25,29);(9,11)(10,13)(12,16)(14,15)(17,19,23,31)(18,25,24,29)(20,28,32,21)(22,30,27,26);(2,6)(3,8)(10,14)(13,15)(17,20,23,32)(18,26,24,30)(19,21,31,28)(22,25,27,29);(1,3)(2,7)(4,6)(5,8)(9,10)(11,13)(12,14)(15,16)(17,21)(18,27)(19,20)(22,24)(23,28)(25,26)(29,30)(31,32);(1,4,5,7)(2,8,6,3)(9,11)(10,15)(12,16)(13,14)(17,22,23,27)(18,28,24,21)(19,30,31,26)(20,25,32,29)Polycyclic presentation:
2^9|a^2=h,c^2=h,d^2=h,f^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, 10125756);