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 | |
4936 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)(11,15)(12,13)(14,16)(17,18,23,22)(19,27,29,24)(20,30,31,25)(21,32,28,26);(1,2)(3,4)(5,6)(7,8)(10,13)(14,15)(17,19,23,29)(18,24,22,27)(20,28,31,21)(25,26,30,32);(2,6)(4,7)(9,10)(11,14)(12,13)(15,16)(17,20,23,31)(18,25,22,30)(19,28,29,21)(24,26,27,32);(2,6)(3,8)(9,11)(10,14)(12,16)(13,15)(17,21)(18,26)(19,20)(22,32)(23,28)(24,30)(25,27)(29,31);(1,3)(2,7)(4,6)(5,8)(17,19)(18,24)(20,28)(21,31)(22,27)(23,29)(25,26)(30,32);(1,4,5,7)(2,3,6,8)(17,22,23,18)(19,27,29,24)(20,25,31,30)(21,32,28,26)Polycyclic presentation:
2^9|a^2=h,b^2=h,c^2=h,f^2=gi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dghi,e^b=egi,f^c=fgi,e^d=egi92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8305856);